Tree representations of graphs

A graph is chordal if and only if it is the intersection graph of some family of subtrees of a tree. Applying “tolerance” allows larger families of graphs to be represented by subtrees. A graph G is in the family [∆, d, t] if there is a tree with maximum degree ∆ and subtrees corresponding to the vertices of G such that each subtree has maximum degree at most d and vertices of G are adjacent if and only if the subtrees corresponding to them have at least t common vertices. It is known that both [3, 3, 1] and [3, 3, 2] are equal to the family of chordal graphs. Furthermore, one can easily observe that every graph G belongs to [3, 3, t] for some t. Denote by t(G) the minimum t so that G ∈ [3, 3, t]. In this paper, we study t(G) and parameters t(n) = min{t : G ∈ [3, 3, t] for every G ⊆ Kn} and tbip(n) = min{t : G ∈ [3, 3, t] for every G ⊆ Kn,n}. Preprint submitted to Elsevier Science 19 April 2006 In particular, our results imply that log n < tbip(n) ≤ 5n1/3 log2 n and log(n/2) < t(n) ≤ 20n1/3 log2 n. 1 Intersection representations of graphs One of important and interesting topics in graph theory is the representation of a graph using intersections of finite sets. Here, each vertex of a given graph is assigned a finite set, and two vertices are adjacent if and only if the corresponding sets intersect. More generally, a p-intersection representation of a graph G with vertices v1, v2, . . . , vn is a collections of sets {S1, S2, . . . , Sn} such that vivj is an edge of G if and only if |Si∩Sj| ≥ p. The p-intersection number θp(G) of G is the smallest cardinality of ⋃n i=1 Si, taken over all p-intersection representations {S1, S2, . . . , Sn} of G. Erdős, Goodman, and Pósa [7] proved that for all G on n vertices, the intersection number of G, θ1(G) is at most n /4. For p > 1, p-intersection numbers have been studied [3,5,6,8], yet many questions remain open. Since each graph has an intersection representation, we can impose additional restrictions on sets allowed in the intersection representation and investigate what families can be obtained. The best known example is the family of interval graphs for which we are allowed to choose only sets that are intervals on the real line or, alternatively, subpaths of a path. This is further generalized in the following definition. Definition 1 For three positive integers ∆, d, and t, we say that a graph G has a (∆, d, t)-representation (and write G ∈ [∆, d, t]) if the following is true. There exists a tree T with maximum degree ∆(T ) ≤ ∆ for which there are subtrees T1, . . . , Tn such that (a) ∆(Ti) ≤ d for every i = 1, . . . , n, (b) vivj ∈ E(G) if and only if |V (Ti) ∩ V (Tj)| ≥ t for all 1 ≤ i < j ≤ n. 1 The author was supported in part by the Hungarian National Science Foundation under grant OTKA NK 62321 and by the National Science Foundation under grant DMS-0140692. 2 The author was partially supported by National Science Foundation grants DMS0099608 and DMS-0400498. 3 The author was partially supported by National Science Foundation grant INT-0305793, by National Security Agency grant H98230-04-1-0035, by CNPq (Proc. 479882/2004–5), and by FAPESP (Proj. Temático–ProNEx Proc. FAPESP 2003/09925–5 and Proc. FAPESP 2004/15397–4).