We investigate which chordal graphs have a representation as intersection graphs of pseudosegments. The main contribution is a construction which shows that all chordal graphs which have a representation as intersection graph of subpaths on a tree are representable. A family of intersection graphs of substars of a star is used to show that not all chordal graphs are representable by pseudosegme...

Deciding whether a collection of unrooted trees is compatible is a fundamental problem in phylogenetics. Two different graph-theoretic characterizations of tree compatibility have recently been proposed. In one of these, tree compatibility is characterized in terms of the existence of a specific kind of triangulation in a structure known as the display graph. An alternative characterization exp...

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 network e...

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 ...

We solve the subgroup intersection problem (SIP) for any RAAG G of Droms type (i.e., with defining graph not containing induced squares or paths of length 3): there is an algorithm which, given finite sets of generators for two subgroups H,K 6 G, decides whether H ∩ K is finitely generated or not, and, in the affirmative case, it computes a set of generators for H ∩ K. Taking advantage of the r...

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...