The partition of graphs into nice subgraphs is a central algorithmic problem with strong ties to matching theory. We study the partitioning of undirected graphs into stars, a problem known to be NP-complete even for the case of stars on three vertices. We perform a thorough computational complexity study of the problem on subclasses of perfect graphs and identify several polynomial-time solvabl...

Boolean-width is a recently introduced graph parameter. Many problems are fixed parameter tractable when parametrized by boolean-width, for instance "Minimum Weighted Dominating Set" (MWDS) problem can be solved in O∗(23k) time given a boolean-decomposition of width k, hence for all graph classes where a boolean-decomposition of width O(log n) can be found in polynomial time, MWDS can be solved...

Let G = (V,E) be a finite undirected graph without loops and multiple edges. A subset M ⊆ E of edges is a dominating induced matching (d.i.m.) in G if every edge in E is intersected by exactly one edge of M . In particular, this means that M is an induced matching, and every edge not in M shares exactly one vertex with an edge in M . Clearly, not every graph has a d.i.m. The Dominating Induced ...

The Cops and Robbers game is played on undirected graphs where a group of cops tries to catch a robber. The game was defined independently by Winkler-Nowakowski and Quilliot in the 1980s and since that time has been studied intensively. Despite of that, its computation complexity is still an open question. In this paper we prove that computing the minimum number of cops that can catch a robber ...

A complete graph is the graph in which every two vertices are adjacent. For a graph G = (V,E), the complete width of G is the minimum k such that there exist k independent sets Ni ⊆ V , 1 ≤ i ≤ k, such that the graph G obtained from G by adding some new edges between certain vertices inside the sets Ni, 1 ≤ i ≤ k, is a complete graph. The complete width problem is to decide whether the complete...

The Cops and Robbers game as originally defined independently by Quilliot and by Nowakowski andWinkler in the 1980s has beenmuch studied, but very few results pertain to the algorithmic and complexity aspects of it. In this paper we prove that computing the minimumnumber of cops that are guaranteed to catch a robber on a given graph is NP-hard and that the parameterized version of the problem i...

The Connected Vertex Cover problem is to decide if a graph G has a vertex cover of size at most $k$ that induces a connected subgraph of $G$. This is a well-studied problem, known to be NP-complete for restricted graph classes, and, in particular, for $H$-free graphs if $H$ is not a linear forest (a graph is $H$-free if it does not contain $H$ as an induced subgraph). It is easy to see that Con...

Peg solitaire is a single-player board game. The goal of the game to remove all but one peg from board. on graphs played arbitrary graphs. A graph called solvable if there exists some vertex s such that it possible starting with as initial hole. In this paper, we prove NP-complete decide or not.

The partition of graphs into “nice” subgraphs is a central algorithmic problem with strong ties to matching theory. We study the partitioning of undirected graphs into same-size stars, a problem known to be NPcomplete even for the case of stars on three vertices. We perform a thorough computational complexity study of the problem on subclasses of perfect graphs and identify several polynomial-t...