Adjacency. Adjacency Two vertices u and v are adjacent if there is an edge connecting them. This is sometimes written as u v.

Transcription

1 Terminology

2 Adjeny Adjeny Two verties u nd v re djent if there is n edge onneting them. This is sometimes written s u v. v v is djent to nd ut not to. 2 / 27

3 Neighourhood Neighourhood The open neighourhood N(v) = { u V u v, u v} of vertex v is the set of verties djent to v (not inluding v). The losed neighourhood N[v] = N(v) {v} inludes v. v N(v) = {, } N[v] = {v,, } 3 / 27

4 Degree Degree The degree deg(v) of vertex v is the numer of inident edges. Note tht the degree is not neessrily equl to the rdinlity of neighours. v deg(v) = 3 deg() = 1 deg() = 5 deg() = 1 4 / 27

5 Minimum nd Mximum Degree Minimum nd Mximum Degree For grph G = (V, E), δ(g) denotes the ninimum nd (G) denotes the mximum degree of G, i. e., δ(g) := min{ deg(v) v V } nd (G) := mx{ deg(v) v V }. v δ(g) = 1 (G) = 3 5 / 27

6 Isolted Vertex Isolted Vertex A vertex v is lled isolted, if it hs no neighours, i. e., N(v) =. v The vertex v is n isolted vertex. 6 / 27

7 Universl Vertex Universl Vertex A vertex v is lled universl, if it djent to ll other verties in the grph, i. e., N[v] = V. v The vertex v is universl vertex. 7 / 27

8 Pendnt Vertex Pendnt Vertex A vertex v is lled pendnt, if it djent to extly one other vertex, i. e., N(v) = 1. v The vertex is pendnt vertex. 8 / 27

9 Pth Pth A set P = {v 0, v 1,..., v k } of distint verties is lled pth (of length k) if v i is djent to v i+1 for ll i with 0 i < k. d e f h g i P = {h, e,,, f, g, i} is pth of length 6. 9 / 27

10 Cyle Cyle A pth P = {v 0, v 1,..., v k } is lled yle (of length k + 1) if v 0 is djent to v k. d e f h g i {h, e,,, f, g} is yle of length / 27

11 Chord Chord A hord in pth (or yle) is n edge onneting two non-onseutive verties of the pth (or yle). d e f h g i The edges g, g, nd eg re hords of the yle {h, e,,, f, g}. 11 / 27

12 Indued Pth / Cyle Indued Pth / Cyle A pth (or yle) is lled indued if it hs no hords. For eh k 3, n indued pth of k verties is lled P k nd n indued yle of length k is lled C k. P 3 C 5 12 / 27

13 Distne Distne The distne d(u, v) of two verties u nd v is the length of the shortest pth from u to v. u v d(u, v) = 3 13 / 27

14 Eentriity Eentriity The eentriity e(v) of vertex v is its mximl distne to ny vertex, i. e., e(v) = mx u V d(u, v). u e(u) = 3 14 / 27

15 Eentriity Eentriity The eentriity e(v) of vertex v is its mximl distne to ny vertex, i. e., e(v) = mx u V d(u, v) v 1 1 e(v) = 2 14 / 27

16 Rdius nd Dimeter Dimeter The dimeter dim(g) of grph G is the mximl eentriity of ll verties in G, i. e., dim(g) = mx v V e(v). Rdius The rdius rd(g) of grph G is the miniml eentriity of ll verties in G, i. e., rd(g) = min v V e(v). Lemm For eh grph G, rd(g) dim(g) 2 rd(g) 15 / 27

17 Intervl Intervl The intervl I(u, v) of two verties u nd v is the set of verties whih re on shortest pth from u to v. Formlly, I(u, v) = { w d(u, v) = d(u, w) + d(w, v) }. d e f h g i I(e, d) = {,, d, e, f, g} 16 / 27

18 Intervl Intervl The intervl I(u, v) of two verties u nd v is the set of verties whih re on shortest pth from u to v. Formlly, I(u, v) = { w d(u, v) = d(u, w) + d(w, v) }. d e f h g i I(h, d) = {, d, f, g, h} 16 / 27

19 Projetion Projetion For vertex v nd vertex set S, the projetion Pr(v, S) is the set of verties in S with miniml distne to v. Formlly, Pr(v, S) = { u S d(u, v) = d(v, S) }. d e f h g i Pr(, I(h, d)) = {, g, h} 17 / 27

20 Complement Complement The omplement G = (V, E) of grph G = (V, E) is the grph with the edges not ontined in G, i. e., E = { uv uv / E }. G G 18 / 27

21 Sugrph Sugrph A grph G = (V, E ) is sugrph of grph G = (V, E) if V V nd E E. G G Note tht u, v V V nd uv E does not imply uv E. 19 / 27

22 Indued Sugrph Indued Sugrph For grph G = (V, E) nd set U V, the indued sugrph G[U ] of G is defined s G[U ] = (U, E ) with E = { uv u, v U ; uv E } d d e f e f h g G i h g i G [ {,, d, e, f, g} ] 20 / 27

23 Conneted Component Conneted Component A onneted omponent of n (undireted) grph is mximl sugrph in whih ny two verties n e onneted y pth. A grph with three onneted omponents. 21 / 27

24 Strongly Conneted Component Strongly Conneted Component A direted grph is strongly onneted if every vertex is rehle from every other vertex. A strongly onneted omponent is mximl sugrph whih is strongly onneted. A grph with three strongly onneted omponents. 22 / 27

25 Seprtor Seprtor A vertex set S is lled sepertor of grph G if removing S from G inreses the numer of onneted omponents. u v The set S = {u, v} is seprtor for the given grph. 23 / 27

26 Artiultion Point Artiultion Point A vertex v is n rtiultion point (lso lled ut vertex) if {v} is seprtor, i. e., removing it inreses the numer of onneted omponents. A grph with two rtiultion points. 24 / 27

27 Bridge Bridge An edge is lled ridge if removing it from the grph (while keeping the verties) inreses the numer of onneted omponents. A grph with ridge. 25 / 27

28 Blok Blok A lok (lso lled 2-onneted omponent) is mximl sugrph without rtiultion points. A grph with four loks. 26 / 27