# The Complexity of Helly-$B_{1}$ EPG Graph Recognition

## TL;DR

This paper investigates the computational complexity of recognizing Helly-$B_k$-EPG graphs, proving NP-completeness for the Helly-$B_1$ case and establishing the problem's NP membership for bounded $k$.

## Contribution

It introduces the Helly-$B_k$-EPG recognition problem, proves NP-completeness for Helly-$B_1$-EPG graphs, and shows NP membership for all $k$ polynomially bounded by graph size.

## Key findings

- Helly-$B_k$-EPG recognition is in NP for polynomially bounded $k$.
- Recognizing Helly-$B_1$-EPG graphs is NP-complete.
- NP-completeness persists even for 2-apex and 3-degenerate graphs.

## Abstract

Golumbic, Lipshteyn, and Stern defined in 2009 the class of EPG graphs, the intersection graph class of edge paths on a grid. An EPG graph $G$ is a graph that admits a representation where its vertices correspond to paths in a grid $Q$, such that two vertices of $G$ are adjacent if and only if their corresponding paths in $Q$ have a common edge. If the paths in the representation have at most $k$ bends, we say that it is a $B_k$-EPG representation. A collection $C$ of sets satisfies the Helly property when every sub-collection of $C$ that is pairwise intersecting has at least one common element. In this paper, we show that given a graph $G$ and an integer $k$, the problem of determining whether $G$ admits a $B_k$-EPG representation whose edge-intersections of paths satisfy the Helly property, so-called Helly-$B_k$-EPG representation, is in NP, for every $k$ bounded by a polynomial function of $|V(G)|$. Moreover, we show that the problem of recognizing Helly-$B_1$-EPG graphs is NP-complete, and it remains NP-complete even when restricted to 2-apex and 3-degenerate graphs.