A Polynomial Delay Algorithm for Enumerating Minimal Dominating Sets in Chordal Graphs

An output-polynomial algorithm for the listing of minimal dominating sets in graphs is a challenging open problem and is known to be equivalent to the well-known Transversal problem which asks for an output-polynomial algorithm for listing the set of minimal hitting sets in hypergraphs. We give a polynomial delay algorithm to list the set of minimal dominating sets in chordal graphs, an important and well-studied graph class where such an algorithm was open for a while.Comment: 13 pages, 1 figure, submitte

Computing Minimum Rainbow and Strong Rainbow Colorings of Block Graphs

A path in an edge-colored graph $G$ is rainbow if no two edges of it are colored the same. The graph $G$ is rainbow-connected if there is a rainbow path between every pair of vertices. If there is a rainbow shortest path between every pair of vertices, the graph $G$ is strongly rainbow-connected. The minimum number of colors needed to make $G$ rainbow-connected is known as the rainbow connection number of $G$, and is denoted by $\text{rc}(G)$. Similarly, the minimum number of colors needed to make $G$ strongly rainbow-connected is known as the strong rainbow connection number of $G$, and is denoted by $\text{src}(G)$. We prove that for every $k \geq 3$, deciding whether $\text{src}(G) \leq k$ is NP-complete for split graphs, which form a subclass of chordal graphs. Furthermore, there exists no polynomial-time algorithm for approximating the strong rainbow connection number of an $n$-vertex split graph with a factor of $n^{1/2-\epsilon}$ for any $\epsilon > 0$ unless P = NP. We then turn our attention to block graphs, which also form a subclass of chordal graphs. We determine the strong rainbow connection number of block graphs, and show it can be computed in linear time. Finally, we provide a polynomial-time characterization of bridgeless block graphs with rainbow connection number at most 4.Comment: 13 pages, 3 figure

The VC-Dimension of Graphs with Respect to k-Connected Subgraphs

We study the VC-dimension of the set system on the vertex set of some graph which is induced by the family of its $k$-connected subgraphs. In particular, we give tight upper and lower bounds for the VC-dimension. Moreover, we show that computing the VC-dimension is $\mathsf{NP}$-complete and that it remains $\mathsf{NP}$-complete for split graphs and for some subclasses of planar bipartite graphs in the cases $k = 1$ and $k = 2$. On the positive side, we observe it can be decided in linear time for graphs of bounded clique-width

Enumerating Minimal Connected Dominating Sets in Graphs of Bounded Chordality

Listing, generating or enumerating objects of specified type is one of the principal tasks in algorithmics. In graph algorithms one often enumerates vertex subsets satisfying a certain property. We study the enumeration of all minimal connected dominating sets of an input graph from various graph classes of bounded chordality. We establish enumeration algorithms as well as lower and upper bounds for the maximum number of minimal connected dominating sets in such graphs. In particular, we present algorithms to enumerate all minimal connected dominating sets of chordal graphs in time O(1.7159^n), of split graphs in time O(1.3803^n), and of AT-free, strongly chordal, and distance-hereditary graphs in time O^*(3^{n/3}), where n is the number of vertices of the input graph. Our algorithms imply corresponding upper bounds for the number of minimal connected dominating sets for these graph classes

Defensive alliances in graphs: a survey

A set $S$ of vertices of a graph $G$ is a defensive $k$-alliance in $G$ if every vertex of $S$ has at least $k$ more neighbors inside of $S$ than outside. This is primarily an expository article surveying the principal known results on defensive alliances in graph. Its seven sections are: Introduction, Computational complexity and realizability, Defensive $k$-alliance number, Boundary defensive $k$-alliances, Defensive alliances in Cartesian product graphs, Partitioning a graph into defensive $k$-alliances, and Defensive $k$-alliance free sets.Comment: 25 page