## Bornstein, Claudson F. and Golumbic, Martin Charles and Santos, Tanilson D. and Souza, Uéverton S. and Szwarcfiter, Jayme L. - The Complexity of Helly-$B_{1}$ EPG Graph Recognition

dmtcs:5603 - Discrete Mathematics & Theoretical Computer Science, June 4, 2020, vol. 22 no. 1 - https://doi.org/10.23638/DMTCS-22-1-19
The Complexity of Helly-$B_{1}$ EPG Graph Recognition

Authors: Bornstein, Claudson F. and Golumbic, Martin Charles and Santos, Tanilson D. and Souza, Uéverton S. and Szwarcfiter, Jayme L.

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.

Volume: vol. 22 no. 1
Section: Graph Theory
Published on: June 4, 2020
Submitted on: June 27, 2019
Keywords: Computer Science - Discrete Mathematics,Computer Science - Computational Complexity,Computer Science - Data Structures and Algorithms