Vertex intersection graphs of paths on a grid: characterization within block graphs

TL;DR

This paper characterizes $B_0$-VPG graphs within block graphs, providing a forbidden subgraph characterization and an alternative recognition algorithm for this class.

Contribution

It offers a minimal forbidden induced subgraph characterization of $B_0$-VPG graphs restricted to block graphs and presents a certifying recognition algorithm.

Findings

Characterization of $B_0$-VPG graphs within block graphs

Forbidden induced subgraph characterization

Alternative certifying recognition algorithm

Abstract

We investigate graphs that can be represented as vertex intersections of horizontal and vertical paths in a grid, the so called $B_0$-VPG graphs. Recognizing this class is an NP-complete problem. Although, there exists a polynomial time algorithm for recognizing chordal $B_0$-VPG graphs. In this paper, we present a minimal forbidden induced subgraph characterization of $B_0$-VPG graphs restricted to block graphs. As a byproduct, the proof of the main theorem provides an alternative certifying recognition and representation algorithm for $B_0$-VPG graphs in the class of block graphs.