Abstract. We present results related to the vertex degree in a uniform s-intersection graph which has received much interest recently. Specifically, we derive the probability distribution for the minimum vertex degree, and show that the number of vertices with an arbitrary degree converges to a Poisson distribution. A uniform s-intersection graph models the topology of a secure wireless sensor ...

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...

For a set of non-negative integers L, the L-intersection number of a graph is the smallest number l for which there is an assignment of subsets Av ⊆ {1, . . . , l} to vertices v, such that every two vertices u, v are adjacent if and only if |Au ∩ Av| ∈ L. The bipartite L-intersection number is defined similarly when the conditions are considered only for the vertices in different parts. In this...

A graph is edge-distance-regular when it is distance-regular around each of its edges and it has the same intersection numbers for any edge taken as a root. In this paper we give some (combinatorial and algebraic) proofs of the fact that every edge-distance-regular graph Γ is distance-regular and homogeneous. More precisely, Γ is edge-distance-regular if and only if it is bipartite distance-reg...

A graph is balanced if its clique-vertex incidence matrix is balanced, i.e., it does not contain a square submatrix of odd order with exactly two ones per row and per column. Interval graphs, obtained as intersection graphs of intervals of a line, are well-known examples of balanced graphs. A circular-arc graph is the intersection graph of a family of arcs on a circle. Circular-arc graphs gener...

Let S be a semigroup. The intersection graph of an S-act A, denoted by G(A), is the undirected simple graph obtained by setting all non-trivial subacts of A to be the vertices and defining two distinct vertices to be adjacent if and only if their intersection is non-empty. It is investigated the interplay between the algebraic properties of A and the graph-theoretic properties of G(A). Also som...

Consider a family of sets and a single set, called query set. How can one quickly find a member of the family which has a maximal intersection with the query set? Strict time constraints on the query and on a possible preprocessing of the set family make this problem challenging. Such maximal intersection queries arise in a wide range of applications, including web search, recommendation system...

Motivated by Chudnovsky’s structure theorem of bull-free graphs, Abu-Khzam, Feghali, and Müller have recently proved that deciding if a graph has a vertex partition into disjoint cliques and a trianglefree graph is NP-complete for five graph classes. The problem is trivial for the intersection of these five classes. We prove that the problem is NP-complete for the intersection of two subsets of...

Let G be the intersection graph of a finite family of convex sets obtained by translations of a fixed convex set in the plane. We show that every such graph with clique number k is (3k − 3)-degenerate. This bound is sharp. As a consequence, we derive that G is (3k − 2)-colorable. We show also that the chromatic number of every intersection graph H of a family of homothetic copies of a fixed con...