Graphs Representable by Caterpillars. Nancy Eaton

Transcription

1 Graphs Representable by Caterpillars Nancy Eaton Definition 1. A caterpillar is a tree in which a single path (the spine) is incident to (or contains) every edge. 1

2 2 Graphs Representable by Caterpillars Outline (1) Definitions (2) Host - Trees in general (3) Host - Path (4) Various questions (5) Host - Subdivision of K 1,3 (6) Host - Caterpillar

3 Tree representation of graphs 3 Definition 2. Given a tree T and a graph G, a T -representation of G is an assignment u S u of subtrees of T to the vertices of G so that {u, v} E(G) V (S u ) V (S v ). The tree T is called the host tree and a graph that has a tree representation is called a subtree graph. Definition 3. Given a set of labels S and a graph G we say that the assignment of subsets of S to the vertices of G, v S v is a p-intersection representation of G if {u, v} E(G) V (S u ) V (S v ) p. θ p (G) is the minimum S, taken over all possible p-intersection representations of G.

4 4 Example: Tree representations of G a G a b c c b c d d Host Tree a b c d

5 Tree representations of graphs 5 Question 1. Do all graphs have tree representations? Ans: No. A B D C C 4 is not a subtree graph. S A and S C must be disjoint with a unique path, within the host, joining them. This unique path is contained in any subtree which has a non-empty intersection with both S A and S C. Therefore, S B and S D cannot be disjoint.

6 6 Tree representations of graphs An early result in this area, is due separately to Buneman 74, Gavril 74, and Walter 78. Theorem 1. A graph is a subtree graph it is a chordal graph. Definition 4. A graph is chordal if it has no induced subgraph isomorphic to a cycle, C n where n 4.

7 Tree representations of graphs 7 Improvement on this result: Theorem 2. (McMorris & Scheinerman, 91) G is chordal there exists a T representation of G where (T ) 3.

8 8 Tree tolerance representations of graphs Definition 5. Given a host tree T and a graph G we say that the assignment of subtrees of T to the vertices of G, v S v is a T representation of G with tolerance t if {u, v} E(G) V (S u ) V (S v ) t Question 2. Does every graph have a tree tolerance representation for some tolerance at least 2? Ans: Yes. We notice that this can be accomplished with a host tree of maximum degree θ 2 (G), which could be quite large.

9 Tree tolerance representations of graphs 9 a a G c d b c c d c f b c e e f Host Tree a b c d e f A tree tolerance representation of G, with tolerance=2

10 10 Path representations The host tree is a PATH. Fact 1. A graph G has a path representation with tolerance t if and only if it has a path representation with tolerance t + 1. Question 3. Which graphs are path representable. Ans: Completely characterized.

11 Path representations 11 Definition 6. A set of vertices in a graph {x 1, x 2, x 3 } is called an asteroidal triple if for each pair {i, j} there is an x i, x j -path that the third vertex is not adjacent to. A graph which contains an asteroidal triple is called an asteroidal graph. x 1 x 1 x 2 x 2 x 3 x 3 Two asteroidal graphs

12 12 Path representations The following theorem was proved by Lekkerkerker and Boland in 67. Theorem 3. G is representable by a path G is chordal and not asteroidal. Notice that we do happen to have a T -representation of an asteroidal graph on host tree T = S 3.

13 Asteroidal triple 13 a G a b c c b c d d Host Tree a b c d

14 14 Some questions There are many open questions that are interesting to consider. Question 4. Given a graph G and positive integer t, what is the minimum positive integer such that there is a tree T with (T ) = and a T -representation of G with tolerance t? Fact 2. For all graphs G on n vertices and all t 2, n 2 /2.

15 Some questions 15 Question 5. Given a graph G and positive integer, what is the minimum positive integer t such that there is a tree T with (T ) = and a T -representation of G with tolerance t? For = 2, done. If = 3, we know that if G is chordal, then t = 1. What if G is not chordal? Given K 2,2 and = 3, t = 3. Given K 3,3 and = 3, t = 3. Given K 4,4 and = 3, t = 4. (Jamison, Mulder, 00).

16 16 Some questions Theorem 4. (E, Z. Füredi, A. V. Kostochka, J. Skokan) Let t = t(n) be the minimum value such that K n,n is representable by a host tree T of T = 3 with tolerance t. There exists a constant c such that for all n t(n) cn 1/3 log n and for all n, log n t(n). 6

17 Some questions 17 Question 6. Given two positive integers t and, let [, t] be the class of graphs for which there exists a tree T with (T ) = and a T representation of G with tolerance t? Solved: [2, t] for any t. Chordal and not asteroidal. Solved [3, 1]=[3, 2]. Chordal graphs. Theorem 5. (E,F,K,S) [h, t] [h, t + 1]. Thus: [3, 1] = [3, 2] [3, 3] [3, 4]...

18 18 Subdivisions of K 1,3 The host H m is a subdivision of K 1, m If for some m, G is H m -representable with tolerance t, we say G H(t) Theorem 6. (E, M.A. Barbato) H(1) = H(2) and H(t) H(t + 1) Characterization of H(1)?

19 Subdivisions of K 1,3 19 For n 4, C n H(1) = H(2). For t = 3, 4, 5, the maximum n such that C n H(t) is n = 3t 3. And if C n H(t) then C n 1 H(t).

20 20 Subdivisions of K 1,3 Theorem 7. (E, M.A. Barbato) Let t 6 and n = n(t) be maximum such that C n H(t), then 1 4 t2 + t n 1 4 t t 3 4. We showed that the subfamily of cycles in H(t) is the family of induced cycles of the triangular lattice with corners removed and t 1 vertices on each of the 3 sides.

21 Caterpillars - joint work with G. Faubert 21 Definition 7. Let Cat[h, t] represent the family of graphs representable by a caterpillar of maximum degree h with tolerance t. Cycle lengths in Cat[h, t] h\t all all 5 3 h By examining the cycle lengths, we found that some classes are different from others. Also note that Cat[h, t] Cat[h + 1, t] and Cat[h, t] Cat[h, t + 1].

22 22 Caterpillars - joint work with G. Faubert Question 7. For a given n, are there any families, Cat[h(n), t(n)], that contain all graphs on n vertices? Fact 3. (E,F,K,S) For each p 1, G Cat[3, θ p (G) + p], for all G. θ p (G) Fact 4. For each p 1, for all G, G Cat[θ p (G), p + 1]. (Use the star with θ p (G) leaves.) And we know for instance that θ 1 (G) n2 4, for all G on n vertices.

23 Caterpillars - joint work with G. Faubert 23 Question 8. Are there fixed values of h and t, such that Cat[h, t] contains all graphs? This might be possible since, for h 4, Cat[h, 3] contains C n for all n and, for t 4, Cat[3,t] contains C n for all n.

24 24 Caterpillars - joint work with G. Faubert Question 9. Can we find a partition of {Cat[h, t] : h 3, t 1}, so that the members of each class represent the same family of graphs? Theorem 8. Cat[3,1] Cat[2,1]. Cat[3,1] = Cat[3,2] = Cat[h,1], h 3. Cat[4,2] = Cat[3,3]. Cat[h,2] Cat[h 1,3], h 5. h\t

25 Caterpillars - joint work with G. Faubert 25 Definition 8. A simplicial vertex in a graph is one whose neighbors form a clique. We say that G has an asimplicial asteroidal triple if there exits in G an asteroidal triple {x 1, x 2, x 3 } of vertices, none of which are simplicial. x 1 x 1 x 2 x 2 x 3 x 3 Two graphs, each containing an asimplicial asteroidal triple

26 26 Caterpillars - joint work with G. Faubert Our Main Result: Theorem 9. G Cat[3, 1] G is chordal and has no asimplicial asteroidal triples.