The Complexity of Helly-B_1 EPG Graph Recognition
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Golumbic, Lipshteyn and Stern defined in 2009 the class of EPG graphs, as the intersection graph 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 changes of direction (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 the problem of recognizing B_k-EPG graphs G=(V,E) whose edge-intersections of paths in a grid satisfy the Helly property, so-called Helly-B_k EPG graphs, is in NP, for every k bounded by a polynomial of |V(G)|. In addition, we show that recognizing Helly-B_1 EPG graphs is NP-complete, and it remains NP-complete even when restricted to 2-apex and 3-degenerate graphs.
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