episciences.org_5603_1675651012 1675651012 episciences.org raphael.tournoy+crossrefapi@ccsd.cnrs.fr episciences.org Discrete Mathematics & Theoretical Computer Science 1365-8050 06 04 2020 vol. 22 no. 1 Graph Theory The Complexity of Helly-\$B_{1}\$ EPG Graph Recognition 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. 06 04 2020 5603 https://arxiv.org/licenses/nonexclusive-distrib/1.0 arXiv:1906.11185 10.48550/arXiv.1906.11185 https://arxiv.org/abs/1906.11185v2 https://arxiv.org/abs/1906.11185v1 10.23638/DMTCS-22-1-19 https://dmtcs.episciences.org/5603 https://dmtcs.episciences.org/6506/pdf https://dmtcs.episciences.org/6506/pdf