Define digraphs: Digraph Definition & Meaning — Merriam-Webster
Posted ondefinition in the Cambridge English Dictionary
digraph
It should be noted that the control items included a number of words with digraphs, and it was these that provoked typical orthographic errors.
From the Cambridge English Corpus
There was no significant difference in the number of morphologically simple and complex answers given in the digraph group, however, both ts
From the Cambridge English Corpus
In the digraph group, 10 nonwords were used.
From the Cambridge English Corpus
A pointed digraph is a digraph with a choice of an initial vertex.
From the Cambridge English Corpus
We used pairs of words containing the same target vowel, represented by either a singleton vowel letter or a vowel digraph.
From the Cambridge English Corpus
In particular, vowel digraphs consisting of the same vowel letter repeated twice reliably represent long vowels.
From the Cambridge English Corpus
The consonant h is always silent or part of the phonologically stable digraphs ch, lh, and nh.
From the Cambridge English Corpus
A loopless finite relation will be called a digraph.
From the Cambridge English Corpus
We point out that our graphs are usually called directed graphs without multiple arcs or digraphs.
From the Cambridge English Corpus
Vowels spelled with a digraph were 3% longer than those spelled with a singleton letter.
From the Cambridge English Corpus
A given regular language is typically parameterized by infinitely many different digraphs.
From the Cambridge English Corpus
A sequential approach to blending, digraphs, phonics etc, was employed.
From the Cambridge English Corpus
These conjectures were proved by the author for vertex-symmetric digraphs [6].
From the Cambridge English Corpus
Since each node of either digraph has at most one incoming directed edge, we need only show that there are no directed cycles.
From the Cambridge English Corpus
The phonics subtest assesses the recognition of consonant clusters, consonant digraphs, variant single consonants, short vowels, long vowels, and vowel combinations.
From the Cambridge English Corpus
These examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors.
All About The Vowel Team Syllable (Vowel Digraphs & Vowel Diphthongs)
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Continuing with my syllable series, learn what a vowel team syllable is and how to teach it using multisensory methods.
You can find all my syllable posts here. And you can read The 6 Types of Syllables here, where I explain what a syllable is, how to count syllables, and go over the 6 syllable types.
What is a vowel team syllable?
A vowel team syllable, also known as a vowel digraph or diphthong syllable, is a syllable with two vowels working together to make one sound.
It’s important to notice whether a vowel combination is reversed such as io in lion. In this case, you split the syllable between the i and o as this is not a vowel team.
Vowel Digraphs and Vowel Diphthongs
A digraph is when two letters spell one sound, and diphthongs are a special kind of vowel sound. So all vowel teams are digraphs but some are also diphthongs.
Vowel diphthongs are known as sliding sounds. These include oi/ow like in oil/boy and ow/ou like in cow/loud. They still make one sound but it’s not as clear or familiar, as the sound slides from the first vowel into the second. If you look in the mirror as you make a diphthong sound, you’ll notice the shape of your mouth changes.
Here are all the different types of vowel digraphs:
- Long A Vowel Teams: ai, ay, ea, eigh, ey
- Long E Vowel Teams: ee, ea, ey, ei, ie
- Long I Vowel Teams: ie, igh
- Long O Vowel Teams: oa, oe
- Long U Vowel Teams: ew, ue, eu
- Diphthong Vowel Teams: oi, oy, ou, ow, au, aw, oo
Unpredictable Vowel Teams
It’s also important to note that some vowel teams almost always make the same sound, but some make different sounds.
For example, ai almost always makes the long a sound, but ea can make the long e, short e, or long a sound. These are known as unpredictable vowel teams.
Unpredictable vowel teams often follow spelling generalizations so these must be taught explicitly!
Vowel Team Words
You’ll see vowel teams and diphthongs in one syllable and multisyllabic words.
Many one syllable words include vowel teams so students don’t need to know how to do syllable division to start learning about this.
Here are some common vowel team words you can use to introduce the vowel team syllable:
You can download my long vowel word lists which include vowel team words for each long vowel sound for free by signing up below.
When should you teach the vowel team syllable?
This syllable is usually taught in first grade but it should be one of the last. Students should already be familiar with open, closed, final e, r-controlled, and consonant -le syllable types.
Teaching Vowel Team Syllables
As always, ensure you have students marking vowels and consonants when learning about syllables. This helps them see the syllable patterns so they can begin to break down words on their own.
Teach students the two vowels in a vowel team work together so they can’t be split up. The 2 vowels act like 1 vowel.
For vowel teams, I don’t have my students mark if it’s a long or short sound (with the breve or macron symbols), since you can’t do either with diphthongs. So I simply have them write a v below each vowel and draw a little scoop underneath that connects them to visualize that the 2 vowels make 1 sound.
Tips for teaching vowel teams:
Introduce vowel digraphs in isolation using phonogram cards or a sound wall.
Teach one vowel team at a time. This is especially important for struggling readers. I would focus on one pattern for at least 2 or 3 days until students can recognize them easily without prompting.
Some students will need a week or longer.
Start with one syllable words then move on to multisyllabic words.
Use lots of visuals like color coding (same color for both vowels in a vowel team), letter tiles, phonogram cards, and posters like the one below.
Use etymology, teach homophones, and teach spelling generalizations to explain the different spelling patterns and rules.
For example, English words do not end in ‘u’ or ‘i’, which explains why some spelling patterns for vowel teams appear in the middle or end of the word.
Some kids get discouraged and confused because some patterns make more than one sound. Be sure to include some content to explain why this is the case so they don’t think “English is hard and doesn’t make sense and has all these exceptions!“
If they can understand why ea makes different sounds depending on the word, that will help them feel more confident in reading and spelling.
Teach vowel diphthongs last, and use mirrors so students can see how their mouth position changes. Focus on the most common vowel teams first like ea, ai, and ee. Diphthongs are a little trickier so save those for last. Using a mirror helps students correctly pronounce the sounds and notice slight differences.
In vowel teams, y and w act as vowels and not consonants. This is why when I introduce vowels I always say sometimes y and w act as vowels too.
I also avoid using that catchy little phrase “When two vowels go walking, the first one does the talking” because it is not always true.
You can download the Vowel Team Syllable Poster (along with the other 5 syllable types) for free in my freebies library.
Vowel Team Syllable Activities
This bundle below includes all my activities for long vowels, which includes vowel teams for each long vowel sound. Learn more about it here.
Color Coding/Tracing/Isolating
One of the first activities I do when teaching a vowel digraph is to isolate that pattern in words with color coding. Provide students with a list of words with vowel teams and use a highlighter to trace the vowel team within the word. From there you can use a short decodable passage and ask them to read and trace all the vowel team syllables they can find.
Some other multisensory writing ideas include air writing, tracing, and sand trays.
Elkonin boxes
Have students use blocks to segment the sounds in words with vowel teams. Say or show the word and have students place a block in each box to represent each sound/phoneme. This helps them visualize vowel teams as one sound even though they contain 2 or more letters.
Students can also tap out the phonemes in a word as they spell it aloud.
Pictured below is my Winter Themed Word Mapping Mats that incorporate sound boxes, segmenting, blending, and writing all on one page.
Blending drills
Use phonogram cards or letter tiles that include vowel teams as one unit (instead of using separate a and e cards to make ea). The cards pictured below are available here.
Do Simultaneous Oral Spelling drills
Use SOS, a multisensory spelling strategy, to practice spelling words with vowel teams. Only use words with the vowel teams students already learned.
Review often by asking students how to spell vowel team sounds
For example, ask students “What says /ee/?” They should write/say all the different ways to spell the long e sound (only the ones they already learned) such as ee, ea, e_e, ei, ie, and ey. I would review this daily until they have learned them all. It only takes a few minutes and can be done orally with the whole class to save time.
Break apart multisyllable words
This would be one of the last activities I do with students once they are pretty strong with the vowel team. I like to write multisyllabic words with vowel teams on flashcards then have students follow the marking and splitting procedures (mark vowels, draw the scoop, split the word, cut along the split).
You can do this with worksheets too.
Don’t forget to make it multisensory!
Remember that for struggling readers the multisensory aspect is critical. Regardless of the activity, incorporate at least 3 senses. If students are tracing, make sure they’re also vocalizing the sound. If students are blending using cards, again they can vocalize the sounds as they blend. You want them to see it, hear it, and write/touch it every time.
Want to remember this? Save All About The Vowel Team Syllable to your favorite Pinterest board!
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Effective Literacy Lesson: Teaching Consonant Digraphs
After students have learned all consonant and short vowel sounds, they are ready to learn consonant digraphs, where two consonants make a special sound when combined.
Professor and Director, Tennessee Reading Research Center
Editor’s Note: This blog post is part of an ongoing series entitled “Effective Literacy Lessons.
” In these posts, we provide a brief summary of the research basis for an approach to teaching reading or writing skills, and an example of how a teacher might “think aloud” to model what students should do in completing portions of the lesson. The intent of these posts is to provide teachers a starting point for designing their own effective literacy lessons.
Research Basis
Regardless of an individual’s age, skilled reading is actually the product of two abilities: decoding and language comprehension (Gough & Tunmer, 1986; Murphy & Farquharson, 2016; Sabatini et al., 2010). These abilities exist in different regions of the brain but must work together in an integrative fashion to support reading comprehension (Aboud, Bailey, Petrill, & Cutting, 2016). If either ability is weak, a student will have difficulty understanding, appreciating, or learning from text.
Decoding instruction focuses on the principles of phonics and proceeds from the easiest skills, such as attributing sounds to individual letters, to the more advanced skills of special sounds made by letter combinations and the position of syllables within words.
In a previous blog post, we identified a common sequence in which students learn how to connect speech sounds (called phonemes) to letters or letter combinations (collectively referred to as graphemes). Digraphs are one of the letter combinations taught after students master single letter sounds. Consonant digraphs are two or more consonants that, together, represent one sound. For example, the consonants “p” and “h” form the grapheme ph that can represent the /f/ sound in words such as “nephew” and “phone.”
This post presents an effective literacy lesson for teaching students to read words with consonant digraphs.
Lesson Materials
For Teachers:
- Scripted purpose, introduction, and modeling
- Plan for guided and independent practice
- Consonant Digraph Cards for the words “chip” and “ship”
For Students:
- Spell the Word
- Pencil
Instructional Sequence
Lesson Appropriate for Kindergarten or Grade 1
This lesson is designed for students who already have learned all consonant sounds and short vowel sounds.
In addition, it is intended for students who have been taught how to blend the sounds of individual letters together to read single-syllable words. Many students in kindergarten are ready to learn consonant digraphs, and all typically developing children should be able to read and spell words with consonant digraphs in first grade. However, students of any age who experience reading difficulties may need explicit instruction in digraphs to continue making progress in their ability to decode words.
1. Establish the Purpose
Teacher script: We have been learning how to read words by saying the sound of each letter and then blending those sounds together to say the word. Let’s read a word together to review.
Display the word “cat.” Ask students to say each letter-sound with you, one at a time and pausing slightly in between each: /c/ /ă/ /t/. Then, slowly blend the sounds together to say the word: “c-a-t.” Finally, quickly blend the sounds to say the word: “cat.
”
You have been doing so well using the letter-sounds to read and spell words! Today, we are going to learn the special sounds that some letters make when they work together. Instead of making the sound of one letter at time, we going to learn when we need to use two letters to make a sound. This will help you be able to read more words that you see in your books.
2. Introduce the Concept
Teacher script: When two consonants work together to make one sound, we call them a digraph. Say that with me: digraph. Let’s say it one more time: digraph. Let’s look at two digraphs.
Display the ch and sh Consonant Digraph Cards (see Supplemental Materials for Teachers).
Refer to each consonant digraph on the cards as you introduce them to the students.
This digraph is made of the letters “c” and “h” but, when we see them together in a word, we won’t sound them out as /k/ /h/. Instead, we will say them as a special sound /ch/.
Say that with me: /ch/. Say that one more time: /ch/. (hold up the ch digraph card) We can remember this digraph with the picture. Who knows what is in this picture?
Students: Chips.
Teacher script: You can see the potato chips in the bag! There are some chips in the picture. The word below the picture is “chip.” Do you hear the sound at the beginning of the word “chip”: /ch/?
This digraph is made of the letters “s” and “h” but, when we see them together in a word, we won’t sound them out as /s/ /h/. Instead, we will say them as a special sound /sh/. Say that with me: /sh/. Say that one more time: /sh/. (hold up the sh digraph card) We can remember this digraph with the picture. Who knows what is in this picture?
Students: Ship.
If a student says “boat,” acknowledge that it is a picture of a boat. Ask if the student knows another word for “boat” that would have the /sh/ sound in it.
Teacher script: Good job thinking of a word with one of our digraphs! This is a picture of a ship. The word below the picture is “ship.” Do you hear the sound at the beginning of the word “ship”: /sh/?
Notice that this digraph is very short: /ch/. We have to stop the sound right away. But we can say the sound of this digraph for a long time: /sh/. (draw out the continuous sound of the digraph) shhhhhhhhhh.
3. Check for Understanding
Teacher script: What is a digraph?
Students: Two consonants that work together to make one sound.
Then, ask students to say the sound for the digraph you show them. Take the two cards and shuffle them in your hands. Randomly show students a card. Provide positive or corrective feedback for their response of /sh/ or /ch/ as appropriate. Continue to shuffle the cards, randomly show a card, and provide feedback. For example, if students provide the incorrect digraph sound, ask them to look for the digraph on the card in the words below the pictures you showed them earlier.
Ask them to say the word and identify the sound of the digraph. Ask them if the sound of the digraph stops right away (i.e., /ch/) or can be continued (i.e., /sh/). If students provide the correct digraph sound, ask them how they remembered the sound for that digraph. Encouraging students to articulate their thinking that led to the correct response has two purposes. First, it will ensure students are not making lucky guesses. Second, hearing peers explain the ways they are remembering the sounds represented by the letters in the digraph will assist any students in the group who may still be struggling to understand or distinguish the digraphs.
Continue shuffling and presenting the cards until students are quickly and accurately responding to the card you display.
4. Introduce the Word Completion Activity
Project the Spell the Word handout (see Supplemental Material for Teachers) on the projection screen.
Teacher script: We are going to practice spelling words with digraphs in them.
We will be using this paper that has pictures on it. Underneath each picture is part of the word that names what is shown in the picture. There are some letters missing! That is why you see a blank line. The part of the word that is missing is the digraph. You will need to decide if the word should be spelled with the ch digraph or the sh digraph. Then, you will write the digraph on the line to finish spelling the word. Sometimes the digraph will be at the beginning of the word, as in our words “chip” and “ship.” The blank line will be before the other letters of the word if the digraph is at the beginning. Sometimes the digraph will be at the end of the word. The blank line will be after the other letters of the word if the digraph is at the end.
5. Model Spelling Words with Digraphs
Teacher script: I am going to show you how to use the picture to decide which digraph you need to complete the spelling of the word.
Refer to the first picture and partially spelled word at the end of the top row as you think aloud.
Teacher script: This is a picture of a man’s face. There is an arrow pointing to his chin. The word I need to spell is “chin.” I’m going to think about the word “chin” and what sounds I recognize in the word: /ch/ /ĭ/ /n/. I can hear the ch digraph, or /ch/ at the beginning of the word “chin.” The ch digraph has the sound that stops right after you say it: /ch/. Now I’m going to look at the word underneath the picture. I see a blank line and then the letters “I» and “n.” I know the letters “I» and “n” make the sounds /ĭ/ /n/. I also know that the blank line before those letters means that the digraph I need is at the beginning of the word. In the word “chin,” the /ch/ sound is at the beginning. I am going to write the letters “c” and “h” on the blank line.
Write the digraph on the handout to complete the spelling.
Now I’m going to blend all the sounds together to read the word and make sure I have the word that names what is in the picture.
Point to the graphemes as you pronounce each: /ch/ /ĭ/ /n/. Then, slowly blend the sounds to say the word: ch-i-n. Finally, quickly blend the sounds to say the word: “chin.”
Teacher script: That matches the picture with the arrow pointing to the man’s chin!
6. Guided Practice Spelling Words with Digraphs
Refer to the first picture and partially spelled word at the start of the second row.
Teacher script: Let’s try the next one together. This is a picture of a woman making a wish as she blows out the candles on her birthday cake. The word we need to spell is “wish.” Everyone say that with me: “wish.” Think about the word “wish” and what sounds you recognize in the word: /w/ /ĭ/ /sh/. Who can tell me which one of our digraphs you hear in the word “wish?”
Students: sh.
Teacher script: You recognized the sound /sh/ and knew that was our sh digraph! Do you hear the sound at the beginning or the end of the word “wish?”
Students: At the end.
Teacher script: When we say the word “wish,” we can hear the /sh/ sound at the end of the word. We can even continue the sound of the sh digraph if we want to: wishhhhhhhh. Now we’re ready to look at the word underneath the picture. Where is the blank line—at the beginning or the end of the word?
Students: At the end.
Teacher script: The blank line is at the end of the word where we hear the digraph in the word “wish.” What letters do you see before the line?
Students: “w” and “i.”
Teacher script: There is a “w” and an “i» before the blank line. Let’s say the sounds that those letters represent: /w/ / ĭ /. Now we need to write our digraph on the line to complete the spelling of the word “wish.” Which letters should I write?
Students: “s” and “h.”
Teacher script: The letters “s” and “h” spell the sound /sh/.
To make sure we have spelled the word that names what is in the picture, let’s blend the sounds together and read the word.
Point to the graphemes as you pronounce each: /w/ /ĭ/ /sh/. Then, slowly blend the sounds to say the word: w-i-sh. Finally, quickly blend the sounds to say the word: “wish.”
Teacher script: That matches the picture of the woman making a birthday wish!
7. Independent Practice Spelling Words with Digraphs
To prepare students to complete the spellings of the remaining words on the handout, it is important to identify what is in each picture. Point to each image on the page and say the word that students will need to spell.
If students are not yet ready to work independently, have them work in pairs to complete the handout. As you monitor individual students or student pairs, refer to their completed words and ask them the following:
- What digraph is in this word?
- What sound does that digraph represent? Is it a sound you have to stop or a sound you can continue?
- How can you remember the sound for this digraph?
- Is the digraph at the beginning or the end of the word? How did you know?
- Say each sound as you point to the letters that spell this word.
Now blend them together and say the word.
Now blend them together and say the word.Asking the questions above will reinforce connecting the sounds and with the appropriate graphemes.
Concluding the Lessons
When students have finished, display samples of their work. Point to different words (not to the pictures) and have the class say the words chorally. Call on individual students to identify the digraphs in selected words and their location in the words.
Conclude the lesson by reviewing all letter-sounds that students know, including the new digraphs that students learned in this lesson. Using cards similar to the samples shown for step 2 (introducing the concept) allows for building a deck of cards cumulatively. This can be a whole-group activity, led by the teacher, or students can use their own versions of the cards for peer practice. The cards should be shuffled, and students should be encouraged to respond quickly with the sound represented by each displayed grapheme. This will build students’ automaticity and thereby reduce the effort of sound-by-sound decoding when they read words in text.
On subsequent days, students can practice reading and spelling the digraphs in different ways such as with word boxes, word sorts, magnetic letters, and digraph word reading fluency.
Supplemental Materials for Teachers
Spell the Word
This handout contains pictures and partially spelled words that contain either the ch or sh digraph. Students will fill in the blank with the correct digraph to complete the spelling for the word that names what is shown in the picture above it.
Consonant Digraph Cards
These cards feature the consonant digraphs ch and sh and can be used as part of a lesson teaching students to read, say, and spell words that have consonant digraphs.
References
Aboud, K. S., Bailey, S. K., Petrill, S. A., & Cutting, L. E. (2016). Comprehending text versus reading words in young readers with varying reading ability: Distinct patterns of functional connectivity from common processing hubs.
Developmental Science, 19, 632-656. https://doi.org/10.1111/desc.12422
Gough, P. B., & Tunmer, W. E. (1986). Decoding, reading, and reading disability. Remedial and Special Education, 7, 6-10. https://doi.org/10.1177/074193258600700104
Murphy, K. A., & Farquharson, K. (2016). Investigating profiles of lexical quality in preschool and their contribution to first grade reading. Reading and Writing: An Interdisciplinary Journal, 29, 1745-1770. https://doi.org/10.1007/s11145-016-9651-y
Sabatini, J. P., Sawaki, Y., Shore, J. R., & Scarborough, H. S. (2010). Relationships among reading skills of adults with low literacy. Journal of Learning Disabilities, 43, 122–138. https://doi.org/10.1177/0022219409359343
Orgraph | it’s… What is a Digraph?
Undirected graph with six vertices and seven edges
In mathematical graph theory and computer science , graph is a collection of objects with links between them.
Objects are represented as vertices , or nodes of a graph, and links as arcs , or edges . For different areas of application, the types of graphs can differ in direction, restrictions on the number of connections, and additional data about vertices or edges.
Many structures of practical interest in mathematics and computer science can be represented by graphs. For example, the structure of Wikipedia can be modeled using a directed graph (digraph), in which the vertices are articles and the arcs (directed edges) are links created by hyperlinks (see Topic Map).
Contents
|
2 Incident matrix Definitions
Graph theory does not have a well-established terminology. In different articles, the same terms mean different things. The definitions below are the most common.
Count
Graph or undirected graph G is an ordered pair G : = ( V , E ) for which the following conditions are satisfied:
-
- V is the set of vertices or nodes ,
- E is a set of pairs (unordered in the case of an undirected graph) of distinct vertices, called edges .
V (and hence E ) are usually considered finite sets. Many good results obtained for finite graphs are wrong (or somehow different) for infinite graphs .
This is because a number of considerations become false in the case of infinite sets.
Vertices and edges of the graph are also called graph elements , the number of vertices in the graph | V | — order , the number of edges | E | — size graph.
Vertices u and v are called terminal vertices (or simply ends ) of edge e = { u , v }. The edge, in turn, connects these vertices. Two end vertices of the same edge are called neighboring .
Two edges are called adjacent if they have a common endpoint.
Two edges are said to be multiples of if the sets of their end vertices are the same.
An edge is called a loop if its ends match, i.e. e = { v , v }.
The degree deg V of the vertex V is the number of edges for which it is the end (the loops are counted twice).
A vertex is called isolated if it is not the end of any edge; hanging (or leaf ) if it is the end of exactly one edge.
Oriented graph
Directed graph (abbreviated as digraph ) G
-
- V is the set of vertices or nodes ,
- A is the set of (ordered) pairs of distinct vertices, called arcs or directed edges .
Arc is an ordered pair of vertices (v, w) , where vertex v is called the beginning and w is called the end of the arc. We can say that the arc v w leads from the vertex v to the vertex w .
Mixed count
Mixed graph G is a graph in which some edges can be directed and some can be undirected. Written as an ordered triple G : = ( V , E , A ), where V , E and A are defined in the same way as above.
It is clear that directed and undirected graphs are special cases of mixed.
Other related definitions
A path (or a chain ) in a graph is a finite sequence of vertices in which each vertex (except the last one) is connected to the next one in the sequence of vertices by an edge.
An oriented path in a digraph is a finite sequence of vertices v i , for which all pairs ( v i , 9007 60076 + 1 ) are (oriented) edges.
A cycle is a path in which the first and last vertices coincide. At the same time, with a length of paths (or cycles) is called the number of its edges . Note that if the vertices u and v are the ends of some edge, then according to this definition, the sequence ( u , v , u ) is a cycle. To avoid such «degenerate» cases, the following notions are introduced.
A path (or cycle) is called simple if no edges are repeated in it; elementary if it is simple and vertices do not repeat in it.
It is easy to see that:
- Any path connecting two vertices contains an elementary path connecting the same two vertices.
- Any simple non-elementary path contains an elementary cycle .
- Every simple cycle passing through some vertex (or edge) contains elementary (sub-)cycle passing through the same vertex (or edge).
The binary relation on the vertex set of the graph, given as «there is a path from u to v «, is an equivalence relation, and therefore partitions this set into equivalence classes, called connectivity components of the graph. If a graph has exactly one connected component, then the graph is connected. On the connected component, one can introduce the notion of the distance between vertices as the minimum length of a path connecting these vertices.
Any maximal connected subgraph of the graph G is called a connected component (or simply a component) of the graph G.
The word «maximum» means maximal with respect to inclusion, that is, not contained in a connected subgraph with a large number of elements
An edge of a graph is called a bridge if its removal increases the number of components.
Additional characteristics of graphs
Count called:
- connected if for any vertices u , v there is a path from u to v .
- strongly connected or oriented connected if it is oriented and there is a oriented path from any vertex to any other.
- tree if it is connected and does not contain simple cycles.
- complete if any two of its (different if loops are not allowed) vertices are connected by an edge.
- Two -bottom , if its vertices can be divided into two unrestrained subset V 1 and V 2 so that any rib connects the vertex of V 1 with a vertex of V
- 3 .
- k-partite if its vertices can be divided into k disjoint subsets0075 k so that there are no edges connecting the vertices of the same subset.
- complete bipartite if each vertex of one subset is connected by an edge to each vertex of another subset.
- planar if the graph can be represented as a diagram on a plane without crossing edges.
- weighted if each edge of the graph is assigned a certain number, called the weight of the edge.
Also happens:
- k-colorable
- k-chromatic
Ways of representing a graph in computer science
Adjacency Matrix
Adjacency matrix is a table where both columns and rows correspond to graph vertices. In each cell of this matrix, a number is written that determines the presence of a connection from the top-row to the top-column (or vice versa).
The disadvantage is the memory requirements — obviously the square of the number of vertices.
Incident matrix
Each row corresponds to a certain vertex of the graph, and the columns correspond to the links of the graph. In the cell at the intersection of i th row with j th column of the matrix is written:
- 1
- if the connection j «leaves» the top i ,
- -1,
- if the connection «enters» the top,
- any number other than 0, 1, −1,
- if the connection is a loop,
- 0
- in all other cases.
This method is the most capacious (the size is proportional to | E | | V | ) and inconvenient for storage, but it makes it easier to find cycles in the graph.
List of edges
Edge list is a type of in-memory graph representation, meaning that each edge is represented by two numbers, the numbers of the vertices of that edge. An edge list is more convenient for implementing various algorithms on graphs than an adjacency matrix.
Generalization of the graph concept
A simple graph is a one-dimensional simplicial complex.
More abstractly, the graph can be defined as a triple , where V and E are some sets ( vertices and edges , respectively), and is the incidence function (or each incidentor) that remaps the incidentor (ordered or unordered) pair of vertices u and v from V (its ends ). Particular cases of this concept are:
- directed graphs (digraphs) — when it is always an ordered pair of vertices;
- undirected graphs — when always an unordered pair of vertices;
- mixed graphs — in which both directed and undirected edges and loops occur;
- Euler graphs — a graph in which there is a cyclic Euler path (Euler cycle).
- multigraphs — graphs with multiples of edges having the same pair of vertices as their ends;
- pseudographs are multigraphs that admit loops;
- simple graphs do not have loops and multiple edges.
Certain other generalizations do not fit the above definition:
- hypergraph — if an edge can connect more than two vertices.
- ultragraph — if there are binary incidence relations between the elements x i and u j .
Literature
- Ore O. Graph Theory. M.: Nauka, 1968. 336s. http://eqworld.ipmnet.ru/ru/library/books/Ore1965ru.djvu
- Wilson R. Introduction to Graph Theory. Per from English. M.: Mir, 1977. 208s. http://eqworld.ipmnet.ru/ru/library/books/Uilson1977ru.djvu
- Harari F. Graph Theory. M.: Mir, 1973. http://eqworld.ipmnet.ru/ru/library/books/Harari1973ru.djvu
- Kormen T. M. and others Part VI. Algorithms for working with graphs // Algorithms: construction and analysis = INTRODUCTION TO ALGORITHMS. — 2nd ed. — M .: «Williams», 2006. — S. 1296. — ISBN 0-07-013151-1
- Salii VN Bogomolov AM Algebraic foundations of the theory of discrete systems. — M .: Physico-mathematical literature, 1997. — ISBN 5-02-015033-9
- Emelichev V. A., Melnikov O. I., Sarvanov V. I., Tyshkevich R. I. Lectures on graph theory. M.: Nauka, 1990. 384 p. (Ed. 2, rev. M.: URSS, 2009. 392 p.)
- Kirsanov M. N. Graphs in Maple. Moscow: Fizmatlit, 2007. — 168 p. http://vuz.exponenta.ru/PDF/book/GrMaple.pdf http://eqworld.ipmnet.ru/ru/library/books/Kirsanov2007ru.pdf
— M .: Physico-mathematical literature, 1997. — ISBN 5-02-015033-9 See also
- Dictionary of Graph Theory
- Direct product of graphs
- Graph theory theorems
- Boost — Library for C++
Links
Popular Graph Visualizers
- aiSee (in Russian)
- David Epstein List (in English)
- List of Roberto Tamassia (in English)
- Georg Sander List (in English)
- Graph Analyzer
Digraph | it’s… What is a Digraph?
Undirected graph with six vertices and seven edges
In mathematical graph theory and computer science , graph is a collection of objects with links between them.
Objects are represented as vertices , or nodes of the graph, and connections — as arcs , or edges . For different areas of application, the types of graphs can differ in direction, restrictions on the number of connections, and additional data about vertices or edges.
Many structures of practical interest in mathematics and computer science can be represented by graphs. For example, the structure of Wikipedia can be modeled using a directed graph (digraph), in which the vertices are articles and the arcs (directed edges) are links created by hyperlinks (see Topic Map).
Contents
|
2 Incident matrix Definitions
Graph theory does not have a well-established terminology. In different articles, the same terms mean different things. The definitions below are the most common.
Count
Graph or undirected graph G is an ordered pair G : = ( V , E ), for which the following conditions are met:
-
- V is the set of vertices or nodes ,
- E is a set of pairs (unordered in the case of an undirected graph) of distinct vertices, called edges .
V (and hence E ) are usually considered finite sets. Many good results obtained for finite graphs are wrong (or different in some way) for infinite counts .
This is because a number of considerations become false in the case of infinite sets.
Vertices and edges of the graph are also called graph elements , the number of vertices in the graph | V | — order , the number of edges | E | — size graph.
Vertices u and v are called terminal vertices (or simply ends ) of edge e = { u , v }. The edge, in turn, connects these vertices. Two end vertices of the same edge are called neighboring .
Two edges are called adjacent if they have a common endpoint.
Two edges are said to be multiples of if the sets of their end vertices are the same.
An edge is called a loop if its ends coincide, i.e. e = { v , v }.
The degree deg V of the vertex V is the number of edges for which it is the end (the loops are counted twice).
A vertex is called isolated if it is not the end of any edge; hanging (or leaf ) if it is the end of exactly one edge.
Oriented graph
Oriented graph (abbreviated as digraph ) G is an ordered pair G : = ( V , A ) for which the following conditions are satisfied:
-
- V is the set of vertices or nodes ,
- A is the set of (ordered) pairs of distinct vertices, called arcs or directed edges .
Arc is an ordered pair of vertices (v, w) , where vertex v is called the beginning and w is called the end of the arc. We can say that the arc v w leads from the vertex v to the vertex w .
Mixed count
Mixed graph G is a graph in which some edges can be directed and some can be undirected.
Written as an ordered triple G : = ( V , E , A ), where V , E and A are defined as above.
It is clear that directed and undirected graphs are special cases of mixed.
Other related definitions
A path (or a chain ) in a graph is a finite sequence of vertices in which each vertex (except the last one) is connected to the next one in the sequence of vertices by an edge.
An oriented path in a digraph is a finite sequence of vertices v i for which all the pairs
A cycle is a path in which the first and last vertices coincide. At the same time, with a length of paths (or cycles) is called the number of its edges . Note that if the vertices u and v are the ends of some edge, then according to this definition, the sequence ( u , v , u ) is a cycle. To avoid such «degenerate» cases, the following notions are introduced.
A path (or cycle) is called simple if no edges are repeated in it; elementary if it is simple and vertices do not repeat in it. It is easy to see that:
- Any path connecting two vertices contains an elementary path connecting the same two vertices.
- Any simple non-elementary path contains an elementary cycle .
- Every simple cycle passing through some vertex (or edge) contains an elementary (sub-)cycle passing through the same vertex (or edge).
The binary relation on the vertex set of the graph, given as «there is a path from u to v «, is an equivalence relation, and therefore partitions this set into equivalence classes, called connectivity components of the graph. If a graph has exactly one connected component, then the graph is connected. On the connected component, one can introduce the notion distances between vertices as the minimum length of a path connecting these vertices.
Any maximal connected subgraph of the graph G is called a connected component (or simply a component) of the graph G. The word «maximum» means maximal with respect to inclusion, that is, not contained in a connected subgraph with a large number of elements
An edge of a graph is called a bridge if its removal increases the number of components.
Additional characteristics of graphs
Count called:
- connected if for any vertices u , v there is a path from u to v .
- strongly connected or oriented connected if it is oriented and there is a oriented path from any vertex to any other.
- tree if it is connected and does not contain simple cycles.
- complete if any two of its (different if loops are not allowed) vertices are connected by an edge.
- Two -bottom , if its vertices can be divided into two unrestrained subset V 1 and V 2 so that any rib connects the vertex of V 1 with a vertex of V
- 3 .
- k-partite if its vertices can be divided into k disjoint subsets0075 k so that there are no edges connecting the vertices of the same subset.
- complete bipartite if each vertex of one subset is connected by an edge to each vertex of another subset.
- planar if the graph can be represented as a diagram on a plane without crossing edges.
- weighted if each edge of the graph is assigned a certain number, called the weight of the edge.
Also happens:
- k-colorable
- k-chromatic
Ways of representing a graph in computer science
Adjacency Matrix
Adjacency matrix is a table where both columns and rows correspond to graph vertices. In each cell of this matrix, a number is written that determines the presence of a connection from the top-row to the top-column (or vice versa).
The disadvantage is the memory requirements — obviously the square of the number of vertices.
Incident matrix
Each row corresponds to a certain vertex of the graph, and the columns correspond to the links of the graph. In the cell at the intersection of i th row with j th column of the matrix is written:
- 1
- if the connection j «leaves» the top i ,
- -1,
- if the connection «enters» the top,
- any number other than 0, 1, −1,
- if the connection is a loop,
- 0
- in all other cases.
This method is the most capacious (the size is proportional to | E | | V | ) and inconvenient for storage, but it makes it easier to find cycles in the graph.
List of edges
Edge list is a type of in-memory graph representation, meaning that each edge is represented by two numbers, the numbers of the vertices of that edge. An edge list is more convenient for implementing various algorithms on graphs than an adjacency matrix.
Generalization of the graph concept
A simple graph is a one-dimensional simplicial complex.
More abstractly, the graph can be defined as a triple , where V and E are some sets ( vertices and edges , respectively), and is the incidence function (or each incidentor) that remaps the incidentor (ordered or unordered) pair of vertices u and v from V (its ends ). Particular cases of this concept are:
- directed graphs (digraphs) — when it is always an ordered pair of vertices;
- undirected graphs — when always an unordered pair of vertices;
- mixed graphs — in which both directed and undirected edges and loops occur;
- Euler graphs — a graph in which there is a cyclic Euler path (Euler cycle).
- multigraphs — graphs with multiples of edges having the same pair of vertices as their ends;
- pseudographs are multigraphs that admit loops;
- simple graphs do not have loops and multiple edges.
Certain other generalizations do not fit the above definition:
- hypergraph — if an edge can connect more than two vertices.
- ultragraph — if there are binary incidence relations between the elements x i and u j .
Literature
- Ore O. Graph Theory. M.: Nauka, 1968. 336s. http://eqworld.ipmnet.ru/ru/library/books/Ore1965ru.djvu
- Wilson R. Introduction to Graph Theory. Per from English. M.: Mir, 1977. 208s. http://eqworld.ipmnet.ru/ru/library/books/Uilson1977ru.djvu
- Harari F. Graph Theory. M.: Mir, 1973. http://eqworld.ipmnet.ru/ru/library/books/Harari1973ru.djvu
- Kormen T. M. and others Part VI. Algorithms for working with graphs // Algorithms: construction and analysis = INTRODUCTION TO ALGORITHMS. — 2nd ed. — M .: «Williams», 2006. — S. 1296. — ISBN 0-07-013151-1
- Salii VN Bogomolov AM Algebraic foundations of the theory of discrete systems. — M .: Physico-mathematical literature, 1997. — ISBN 5-02-015033-9
- Emelichev V. A., Melnikov O. I., Sarvanov V. I., Tyshkevich R. I. Lectures on graph theory. M.: Nauka, 1990. 384 p. (Ed. 2, rev. M.: URSS, 2009. 392 p.)
- Kirsanov M. N. Graphs in Maple. Moscow: Fizmatlit, 2007. — 168 p. http://vuz.exponenta.ru/PDF/book/GrMaple.pdf http://eqworld.ipmnet.ru/ru/library/books/Kirsanov2007ru.pdf
— M .: Physico-mathematical literature, 1997. — ISBN 5-02-015033-9 See also
- Dictionary of Graph Theory
- Direct product of graphs
- Graph theory theorems
- Boost — Library for C++
Links
Popular Graph Visualizers
- aiSee (in Russian)
- David Epstein List (in English)
- List of Roberto Tamassia (in English)
- Georg Sander List (in English)
- Graph Analyzer
Graph Theory Terminology — iRunner Wiki
Contents
- 1 Graphs
- 2 Trees
- 3 Subgraphs
- 4 Chains, loops, paths
- 4. 1 In an undirected graph
- 4.2 In a directed graph 9{(2)}[/math] stands for the set of all two-element subsets (2-combinations) of the set [math]V[/math].
A graph is called empty (or null-graph ) if its edge set is empty. A graph is called full if it contains all possible edges.
The vertex set of the graph [math]G[/math] is denoted by [math]V(G)[/math] or [math]VG[/math],
set of edges — via [math]E(G)[/math] or [math]EG[/math].
The number [math]|V(G)|[/math] of vertices of the graph [math]G[/math] is called its 9{(2)}[/math] edges. The same edges of the multigraph are called multiples of . In other words, a multigraph is a generalization of a graph to the case of multiple edges.Pseudograph is an ordered pair [math](V, E)[/math] from a non-empty set [math]V[/math] of vertices and a family [math]E[/math] of unordered pairs (2-combinations with repetitions ) vertices. The term pseudograph generalizes the notion of a multigraph, allowing for loops and edges that connect a vertex to itself.
92[/math] denotes the set of all ordered pairs (2-arrangements) consisting of two distinct elements [math]V[/math].
The set of arcs of the digraph [math]G[/math] is denoted by [math]A(G)[/math] or [math]AG[/math].
The directed multigraph is defined similarly, with the only difference that the coinciding arcs of the directed multigraph are called parallel .Base of the digraph [math]G[/math] is an undirected multigraph obtained by removing the orientation from the arcs of the digraph [math]G[/math].
Mixed graph is a graph that can contain both arcs and undirected edges.
The term graph may be used instead of any of the generalizations of this concept, if it is clear from the context which definition is meant. Therefore, in order to distinguish the graph in the original definition from others, the concept of a simple graph is used — an undirected graph without loops and multiple edges.
Trees
Tree is a connected graph without cycles.
[1] A directed graph [math]D = (V, A)[/math] is called a directed tree with root [math]r \in V[/math] if each of its vertices is reachable from [math ]r[/math] and the base [math]D_b[/math] of the graph [math]D[/math] is a tree.
Forest (or acyclic graph ) is a graph without cycles. Each component of the forest is a tree. Note that we are only talking about an undirected simple graph here.
Acyclic digraph is a digraph without cycles. It is worth noting that the base of an acyclic digraph may not be an acyclic graph (forest).
Subgraphs
The graph [math]H[/math] is called a subgraph (or part ) of the graph [math]G[/math],
if [math]V(H) \subseteq V(G)[/math] and [math]E(H) \subseteq E(G)[/math].The spanning subgraph (or factor ) is a subgraph containing all the vertices of the original graph.
Frame (or framework ) of the graph [math]G[/math] is an inclusion-maximal forest that is a subgraph of the graph [math]G[/math].
In other words, a spanning tree is a subgraph of the graph [math]G[/math] consisting of one spanning tree for each connected component of the graph [math]G[/math]. It is worth noting that not every spanning forest is a spanning tree, because, for example, an empty spanning subgraph is a forest, but is not a spanning tree if the graph contains at least one edge.
If the set of vertices of the subgraph [math]H[/math] of the graph [math]G[/math] is [math]S[/math], and the subgraph [math]H[/math] itself is maximal (by inclusion) among of all such subgraphs, then the subgraph [math]H[/math] is called subgraph generated by [math]S[/math] , or just generated by subgraph . In other words, a subgraph [math]H[/math] of the graph [math]G[/math] is called generated if it contains all possible (for its set of vertices) edges of the graph [math]G[/math].
Chains, cycles, paths
In an undirected graph
Route — alternating sequence
[math]v_0, e_1, v_1, e_2, \ldots, e_{\ell}, v_{\ell} \tag{1}[/math]
vertices and edges, where [math]e_i = \{\,v_{i-1}, v_i\,\}\label{route}[/math] ([math]i = \overline{1, \ell} [/math]).
The vertices [math]v_0[/math] and [math]v_{\ell}[/math] are called extreme , and all the rest — intermediate (or internal ).
A route containing [math]v_0[/math] and [math]v_{\ell}[/math] as extremes is called [math](v_0, v_{\ell})[/math]-route .If there are no multiple edges in the graph, then the route can be uniquely specified by a sequence of vertices.
The circuit is a route, all edges of which are pairwise distinct.
A simple path is a path all of whose vertices, except possibly the extreme ones, are pairwise distinct.
A chain in a graph can also be considered as a subgraph of this graph. Nevertheless, the subgraph corresponding to a chain defines this chain uniquely (up to direction) if and only if it is simple.
Cyclic route is a route whose extreme vertices coincide.
Cycle (or cyclic circuit ) is a cyclic route that is a circuit.
Simple cycle is a simple cycle circuit.
Hamilton cycle is a simple cycle containing all the vertices of the graph.
Euler cycle is a cycle containing all graph edges.
In a directed graph
A directed route (or simply route ) is a sequence of the form (1) for a directed graph in which [math]e_i = (v_{i — 1}, v_i)[/math].
The concepts of a chain, a cyclic route, and a cycle carry over to the case of a directed graph without change.Path is an oriented route, all vertices of which, except possibly the extreme ones, are different.
Contour — cyclic path.
Half route is a sequence of the form (1) in which [math]e_i = (v_{i-1}, v_i)[/math] or [math]e_i = (v_i, v_{i-1})[ /math]. Similarly, half-circuit , half-path and half-circuit are defined.
If there is a [math](u, v)[/math]-route in a digraph, then we say that the vertex [math]v[/math] is reachable from [math]u[/math] .
Any vertex is considered reachable from itself.
Connectivity
In an undirected graph
A connected graph is a graph in which any two noncoincident vertices are connected by a path.
Connected component (or connected component , or simply component ) of the graph [math]G[/math] is the maximal (by inclusion) connected subgraph of the graph [math]G[/math].
Connected area of a graph is the set of all vertices of one connected component of this graph.
An articulation point is a vertex whose removal increases the number of graph components.
A connected graph that does not contain articulation points is called doubly connected or vertex doubly connected .
A bridge is an edge whose removal increases the number of graph components.
A connected graph without bridges is called edge-to-edge .
In a directed graph
A digraph is called strong (or strongly connected ) if any two of its vertices are reachable from each other.
- 4.
1 In an undirected graph 
[1] 
In other words, a spanning tree is a subgraph of the graph [math]G[/math] consisting of one spanning tree for each connected component of the graph [math]G[/math]. It is worth noting that not every spanning forest is a spanning tree, because, for example, an empty spanning subgraph is a forest, but is not a spanning tree if the graph contains at least one edge.
Any vertex is considered reachable from itself.
