Intersection graphs on trees with a tolerance

An intriguing theme in graph theory is that of the intersection graph of a family of subsets of a set: the vertices of the graph are represented by the subsets of the family and adjacency is defined by a non-empty intersection of the corresponding subsets. Prime examples are interval graphs and chordal graphs. An interval graph is the intersection graph of a family of closed intervals on the real line. A classical result is the characterization of interval graphs by forbidden subgraphs by LEKKERKERKER and BOLAND [7]. A chordal graph is a graph without induced cycles of length four or more. They were proven to be the intersection graphs of a family of subtrees of a host tree by GAVRIL [1] and WALTER [9]. In [8] MCMORRIS and SCHEINERMAN observed, without adding a proof, that this result may be sharpened in the following way: a graph G is chordal if and only if it is the intersection graph of a family of leaf-generated subtrees of a full binary tree such that intersecting subtrees share a leaf. In this very special type of representation 'intersection' may be even replaced by 'edge-intersection'. Note that, if there are no restrictions on the host tree, then any graph is the edge-intersection graphs of some family of subtrees of a tree. Note that one may even take the tree and all subtrees to be stars (of high enough degree). In this paper we study a class of generalizations of chordal graphs using the idea of tolerance intersection graph, cf. [2, 3]. In [2] the idea of tolerance was introduced. And in [3] a very broad Master Plan, based on this idea from [2], was formulated: the study of φ-tolerance (S, μ)-graphs. Let T be a tree of maximum degree Δ, called the host tree, and let {Sv}v  V be a (finite) family of non-empty subtrees of T of maximum degree d, and let t be a positive integer, called the tolerance. Let G = (V, E) be the graph with vertex set V, where uv is an edge if and only if |Su  Sv|  t. Then (T, {Sv}v  V , t) is a {Δ, d, t}-representation of G. A graph G is said to be {Δ, d, t}-representable if it has a {Δ, d, t}-representation. We will denote the class of {Δ, d, t}-representable graphs simply by {Δ, d, t}. This …