B1-EPG representations using block-cutpoint trees

TL;DR

This paper investigates B1-EPG and L-EPG graph classes, showing certain tree subclasses are representable and providing linear-time algorithms for L-EPG representations using block-cutpoint trees.

Contribution

It introduces new techniques based on block-cutpoint trees to characterize and efficiently construct B1-EPG and L-EPG representations for specific graph classes.

Findings

Cactus graphs are B1-EPG.

Block graphs are L-EPG.

Linear-time algorithms for L-EPG representations of generalized trees.

Abstract

In this paper, we are interested in the edge intersection graphs of paths of a grid where each path has at most one bend, called B1-EPG graphs and first introduced by Golumbic et al (2009). We also consider a proper subclass of B1-EPG, the L-EPG graphs, which allows paths only in ``L'' shape. We show that two superclasses of trees are B1-EPG (one of them being the cactus graphs). On the other hand, we show that the block graphs are L-EPG and provide a linear time algorithm to produce L-EPG representations of generalization of trees. These proofs employed a new technique from previous results in the area based on block-cutpoint trees of the respective graphs.