The kk-Dominating Graph

Given a graph $G$, the $k$-dominating graph of $G$, $D_k(G)$, is defined to be the graph whose vertices correspond to the dominating sets of $G$ that have cardinality at most $k$. Two vertices in $D_k(G)$ are adjacent if and only if the corresponding dominating sets of $G$ differ by either adding or deleting a single vertex. The graph $D_k(G)$ aids in studying the reconfiguration problem for dominating sets. In particular, one dominating set can be reconfigured to another by a sequence of single vertex additions and deletions, such that the intermediate set of vertices at each step is a dominating set if and only if they are in the same connected component of $D_k(G)$. In this paper we give conditions that ensure $D_k(G)$ is connected.Comment: 2 figure, The final publication is available at http://link.springer.co

Reconfiguration of Dominating Sets

We explore a reconfiguration version of the dominating set problem, where a dominating set in a graph $G$ is a set $S$ of vertices such that each vertex is either in $S$ or has a neighbour in $S$. In a reconfiguration problem, the goal is to determine whether there exists a sequence of feasible solutions connecting given feasible solutions $s$ and $t$ such that each pair of consecutive solutions is adjacent according to a specified adjacency relation. Two dominating sets are adjacent if one can be formed from the other by the addition or deletion of a single vertex. For various values of $k$, we consider properties of $D_k(G)$, the graph consisting of a vertex for each dominating set of size at most $k$ and edges specified by the adjacency relation. Addressing an open question posed by Haas and Seyffarth, we demonstrate that $D_{\Gamma(G)+1}(G)$ is not necessarily connected, for $\Gamma(G)$ the maximum cardinality of a minimal dominating set in $G$. The result holds even when graphs are constrained to be planar, of bounded tree-width, or $b$-partite for $b \ge 3$. Moreover, we construct an infinite family of graphs such that $D_{\gamma(G)+1}(G)$ has exponential diameter, for $\gamma(G)$ the minimum size of a dominating set. On the positive side, we show that $D_{n-m}(G)$ is connected and of linear diameter for any graph $G$ on $n$ vertices having at least $m+1$ independent edges.Comment: 12 pages, 4 figure

On Kernels, β-graphs, and β-graph Sequences of Digraphs

We begin by investigating some conditions determining the existence of kernels in various classes of directed graphs, most notably in oriented trees, grid graphs, and oriented cycles. The question of uniqueness of these kernels is also handled. Attention is then shifted to $\gamma$-graphs, structures associated to the minimum dominating sets of undirected graphs. I define the $\beta$-graph of a given digraph analogously, involving the minimum absorbant sets. Finally, attention is given to iterative construction of $\beta$-graphs, with an attempt to characterize for what classes of digraphs these $\beta$-sequences terminate

Gamma graphs of some special classes of trees

A set S⊂V is a dominating set of a graph G=(V,E) if every vertex v∈V which does not belong to S has a neighbour in S. The domination number γ(G) of the graph G is the minimum cardinality of a dominating set in G. A dominating set S is a γ-set in G if |S|=γ(G). "Some graphs have exponentially many γ-sets, hence it is worth to ask a question if a γ-set can be obtained by some transformations from another γ-set. The study of gamma graphs is an answer to this reconfiguration problem. We give a partial answer to the question which graphs are gamma graphs of trees. In the second section gamma graphs γ.T of trees with diameter not greater than five will be presented. It will be shown that hypercubes Qk are among γ.T graphs. In the third section γ.T graphs of certain trees with three pendant vertices will be analysed. Additionally, some observations on the diameter of gamma graphs will be presented, in response to an open question, published by Fricke et al., if diam(T(γ))=O(n)