On $B_1$-EPG and EPT graphs

TL;DR

This paper proves that all chordal $B_1$-EPG graphs are also in VPT and EPT classes, and explores structural features of $B_1$-EPG graphs lacking Helly representations, including specific graph families.

Contribution

It establishes the inclusion of chordal $B_1$-EPG graphs in VPT and EPT classes and characterizes structures in $B_1$-EPG graphs without Helly representations.

Findings

Chordal $B_1$-EPG graphs are in VPT and EPT classes.

Structural features of non-Helly $B_1$-EPG graphs identified.

Characterization of specific graph families within Helly-$B_1$ EPG.

Abstract

This research contains as a main result the prove that every Chordal $B_1$-EPG graph is simultaneously in the graph classes VPT and EPT. In addition, we describe structures that must be present in any $B_1$-EPG graph which does not admit a Helly-$B_1$-EPG representation. In particular, this paper presents some features of non-trivial families of graphs properly contained in Helly-$B_1$ EPG, namely Bipartite, Block, Cactus and Line of Bipartite graphs.