On the p-Connectedness of Graphs – a Survey

A graph is said to be p-connected if for every partition of its vertices into two non-empty, disjoint, sets some chordless path with three edges contains vertices from both sets in the partition. As it turns out, p-connectedness generalizes the usual connectedness of graphs and leads, in a natural way, to a unique tree representation for arbitrary graphs. This paper reviews old and new results, both structural and algorithmic, about p-connectedness along with applications to various graph decompositions

On the Enumeration of Minimal Dominating Sets and Related Notions

A dominating set $D$ in a graph is a subset of its vertex set such that each vertex is either in $D$ or has a neighbour in $D$. In this paper, we are interested in the enumeration of (inclusion-wise) minimal dominating sets in graphs, called the Dom-Enum problem. It is well known that this problem can be polynomially reduced to the Trans-Enum problem in hypergraphs, i.e., the problem of enumerating all minimal transversals in a hypergraph. Firstly we show that the Trans-Enum problem can be polynomially reduced to the Dom-Enum problem. As a consequence there exists an output-polynomial time algorithm for the Trans-Enum problem if and only if there exists one for the Dom-Enum problem. Secondly, we study the Dom-Enum problem in some graph classes. We give an output-polynomial time algorithm for the Dom-Enum problem in split graphs, and introduce the completion of a graph to obtain an output-polynomial time algorithm for the Dom-Enum problem in $P_6$-free chordal graphs, a proper superclass of split graphs. Finally, we investigate the complexity of the enumeration of (inclusion-wise) minimal connected dominating sets and minimal total dominating sets of graphs. We show that there exists an output-polynomial time algorithm for the Dom-Enum problem (or equivalently Trans-Enum problem) if and only if there exists one for the following enumeration problems: minimal total dominating sets, minimal total dominating sets in split graphs, minimal connected dominating sets in split graphs, minimal dominating sets in co-bipartite graphs.Comment: 15 pages, 3 figures, In revisio

Graph Algorithms and Applications

The mixture of data in real-life exhibits structure or connection property in nature. Typical data include biological data, communication network data, image data, etc. Graphs provide a natural way to represent and analyze these types of data and their relationships. Unfortunately, the related algorithms usually suffer from high computational complexity, since some of these problems are NP-hard. Therefore, in recent years, many graph models and optimization algorithms have been proposed to achieve a better balance between efficacy and efficiency. This book contains some papers reporting recent achievements regarding graph models, algorithms, and applications to problems in the real world, with some focus on optimization and computational complexity

On interval number in cycle convexity

International audienceRecently, Araujo et al. [Manuscript in preparation, 2017] introduced the notion of Cycle Convexity of graphs. In their seminal work, they studied the graph convexity parameter called hull number for this new graph convexity they proposed, and they presented some of its applications in Knot theory. Roughly, the tunnel number of a knot embedded in a plane is upper bounded by the hull number of a corresponding planar 4-regular graph in cycle convexity. In this paper, we go further in the study of this new graph convexity and we study the interval number of a graph in cycle convexity. This parameter is, alongside the hull number, one of the most studied parameters in the literature about graph convexities. Precisely, given a graph G, its interval number in cycle convexity, denoted by $in_{cc} (G)$, is the minimum cardinality of a set S ⊆ V (G) such that every vertex w ∈ V (G) \ S has two distinct neighbors u, v ∈ S such that u and v lie in same connected component of G[S], i.e. the subgraph of G induced by the vertices in S.In this work, first we provide bounds on $in_{cc} (G)$ and its relations to other graph convexity parameters, and explore its behavior on grids. Then, we present some hardness results by showing that deciding whether $in_{cc} (G)$ ≤ k is NP-complete, even if G is a split graph or a bounded-degree planar graph, and that the problem is W[2]-hard in bipartite graphs when k is the parameter. As a consequence, we obtainthat $in_{cc} (G)$ cannot be approximated up to a constant factor in the classes of split graphs and bipartite graphs (unless P = N P ).On the positive side, we present polynomial-time algorithms to compute $in_{cc} (G)$ for outerplanar graphs, cobipartite graphs and interval graphs. We also present fixed-parameter tractable (FPT) algorithms to compute it for (q, q − 4)-graphs when q is the parameter and for general graphs G when parameterized either by the treewidth or the neighborhood diversity of G.Some of our hardness results and positive results are not known to hold for related graph convexities and domination problems. We hope that the design of our new reductions and polynomial-time algorithms can be helpful in order to advance in the study of related graph problems