We establish some preliminary results for Sutner’s σ game, i.e. “Lights Out,” played on the generalized Petersen graph P (n, k). While all regular Petersen graphs admit game configurations that are not solvable, we prove that every game on the P (2n, n) graph has a unique solution. Moreover, we introduce an exceedingly simple strategy for finding the solution to any game on these graphs. Surpri...

The well-known Disjoint Paths problem is to decide if a graph contains k pairwise disjoint paths, each connecting different terminal pair from set of distinct vertex pairs. We determine, with an exception two cases, the complexity for H -free graphs. If fixed, we obtain -Disjoint problem, which known be polynomial-time solvable on class all graphs every ? 1 . latter does no longer hold need con...

We consider the following vertex-partition problem on graphs: given a graph with real nonnegative edge weights, partition the vertices into clusters (i.e. cliques) to minimize the total weight of edges out of the clusters. This optimization problem is known to be an NP-complete problem even for unweighted graphs and has been studied extensively in the scope of fixed-parameter tractability (FPT)...

In this paper we prove that testing the cyclic level planarity of a cyclic level graph is a polynomial-time solvable problem. This is achieved by introducing and studying a generalization of this problem, which we call cyclic T -level planarity. Moreover, we show a complexity dichotomy for testing the simultaneous level planarity of a set of level graphs, with respect to both the number of leve...

A co-bipartite chain graph is a co-bipartite graph in which the neighborhoods of the vertices in each clique can be linearly ordered with respect to inclusion. It is known that the maximum cut problem (MaxCut) is NP-hard in co-bipartite graphs [3]. We consider MaxCut in co-bipartite chain graphs. We first consider the twin-free case and present an explicit solution. We then show that MaxCut is ...

In 2011, Beeler and Hoilman generalised the game of peg solitaire to arbitrary connected graphs. Since then, related games have been considered on many graph classes. this paper, we introduce a variant solitaire, called path-stick which is played with sticks in edges instead pegs vertices. We prove several analogues results for that game, mainly regarding different Furthermore, characterise, ve...

Interval digraphs were introduced by West et all. They can be recognized in polynomial time and admit a characterization in terms of incidence matrices. Nevertheless, they do not have a forbidden structure characterization nor a low-degree polynomial time recognition algorithm. We introduce a new class of ‘adjusted interval digraphs’, obtained by a slight change in the definition. By contrast, ...

Given an undirected graph on n vertices with weights on its edges, Min WCF(p) consists of computing a covering forest of minimum weight such that each of its tree components contains at least p vertices. It has been proved that Min WCF(p) is NP -hard for any p ≥ 4 (Imielinska et al., 1993) but (2 − 1 n )-approximable (Goemans and Williamson, 1995). While Min WCF(2) is polynomial-time solvable, ...

Finding “good” cycles in graphs is a problem of great interest in graph theory as well as in locational analysis. We show that the center and median problems are NP hard in general graphs. This result holds both for the variable cardinality case (i.e. all cycles of the graph are considered) and the fixed cardinality case (i.e. only cycles with a given cardinality p are feasible). Hence it is of...

Given an undirected graph on n vertices with weights on its edges, Min WCF(p) consists of computing a covering forest of minimum weight such that each of its tree components contains at least p vertices. It has been proved that Min WCF(p) is NP -hard for any p ≥ 4 (Imielinska et al., 1993) but (2− 1 n )-approximable (Goemans andWilliamson, 1995). While Min WCF(2) is polynomial-time solvable, al...