Detecting wheels

A \emph{wheel} is a graph made of a cycle of length at least~4 together with a vertex that has at least three neighbors in the cycle. We prove that the problem whose instance is a graph $G$ and whose question is "does $G$ contains a wheel as an induced subgraph" is NP-complete. We also settle the complexity of several similar problems

On Box-Perfect Graphs

Let $G=(V,E)$ be a graph and let $A_G$ be the clique-vertex incidence matrix of $G$. It is well known that $G$ is perfect iff the system $A_{_G}\mathbf x\le \mathbf 1$, $\mathbf x\ge\mathbf0$ is totally dual integral (TDI). In 1982, Cameron and Edmonds proposed to call $G$ box-perfect if the system $A_{_G}\mathbf x\le \mathbf 1$, $\mathbf x\ge\mathbf0$ is box-totally dual integral (box-TDI), and posed the problem of characterizing such graphs. In this paper we prove the Cameron-Edmonds conjecture on box-perfectness of parity graphs, and identify several other classes of box-perfect graphs. We also develop a general and powerful method for establishing box-perfectness

The Complexity of Helly-B1B_{1} EPG Graph Recognition

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