## Claudson F. Bornstein ; Martin Charles Golumbic ; Tanilson D. Santos ; Uéverton S. Souza ; Jayme L. Szwarcfiter - 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: Claudson F. Bornstein ; Martin Charles Golumbic ; Tanilson D. Santos ; Uéverton S. Souza ; Jayme L. Szwarcfiter

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
Accepted on: June 4, 2020
Submitted on: June 27, 2019
Keywords: Computer Science - Discrete Mathematics,Computer Science - Computational Complexity,Computer Science - Data Structures and Algorithms

 Source : ScholeXplorer HasVersion DOI 10.48550/arxiv.1906.11185 10.48550/arxiv.1906.11185 The Complexity of Helly-$B_{1}$ EPG Graph Recognition Bornstein, Claudson F. ; Golumbic, Martin Charles ; Santos, Tanilson D. ; Souza, Uéverton S. ; Szwarcfiter, Jayme L. ;