On the helly property of some intersection graphs

Abstract

An EPG graph G is an edge-intersection graph of paths on a grid. In this doctoral thesis we will mainly explore the EPG graphs, in particular B1-EPG graphs. However, other classes of intersection graphs will be studied such as VPG, EPT and VPT graph classes, in addition to the parameters Helly number and strong Helly number to EPG and VPG graphs. We will present the proof of NP-completeness to Helly-B1-EPG graph recognition problem. We investigate the parameters Helly number and the strong Helly number in both graph classes, EPG and VPG in order to determine lower bounds and upper bounds for this parameters. We completely solve the problem of determining the Helly and strong Helly numbers, for Bk-EPG, and Bk-VPG graphs, for each value k. Next, we present the result that every Chordal B1-EPG graph is simultaneously in the VPT and EPT graph classes. In particular, we describe structures that occur in B1-EPG graphs that do not support a Helly-B1-EPG representation and thus we define some sets of subgraphs that delimit Helly subfamilies. In addition, features of some non-trivial graph families that are properly contained in Helly-B1 EPG are also presented.