A vertex dominating path in a graph is a path P such that every vertex outside P has a neighbor on P . In 1988 H. Broersma stated a result implying that every n-vertex kconnected graph G such that σ(k+2)(G) ≥ n− 2k − 1 contains a dominating path. We show that every n-vertex k-connected graph with σ2(G) ≥ 2n k+2 + f(k) contains a dominating path of length at most O(|T |), where T is a minimum do...

A subset $S$ of vertex set $V(D)$ is an {em indpendent dominating set} of $D$ if $S$ is both an independent and a dominating set of $D$. The {em indpendent domination number}, $i(D)$ is the cardinality of the smallest independent dominating set of $D$. In this paper we calculate the independent domination number of the { em cartesian product} of two {em directed paths} $P_m$ and $P_n$ for arbi...

An edge dominating set of a graph G = (V,E) is a subset M ⊆ E of edges in the graph such that each edge in E −M is incident with at least one edge in M . In an instance of the parameterized edge dominating set problem we are given a graph G = (V,E) and an integer k and we are asked to decide whether G has an edge dominating set of size at most k. In this paper we show that the parameterized edg...

In this paper, we study a generalization of the paired domination number. Let G= (V ,E) be a graph without an isolated vertex. A set D ⊆ V (G) is a k-distance paired dominating set of G if D is a k-distance dominating set of G and the induced subgraph 〈D〉 has a perfect matching. The k-distance paired domination number p(G) is the cardinality of a smallest k-distance paired dominating set of G. ...

Given a graph G together with a capacity function c : V (G) → N, we call S ⊆ V (G) a capacitated dominating set if there exists a mapping f : (V (G) \ S) → S which maps every vertex in (V (G) \S) to one of its neighbors such that the total number of vertices mapped by f to any vertex v ∈ S does not exceed c(v). In the Planar Capacitated Dominating Set problem we are given a planar graph G, a ca...

For any graph G and a set ~ of graphs, two distinct vertices of G are said to be ~-adjacent if they are contained in a subgraph of G which is isomorphic to a member of ~. A set S of vertices of G is an ~-dominating set (total ~¢~-dominating set) of G if every vertex in V(G)-S (V(G), respectively) is 9¢g-adjacent to a vertex in S. An ~-dominating set of G in which no two vertices are oCf-adjacen...

We give the first linear kernels for Dominating Set and Connected Dominating Set problems on graphs excluding a fixed graph H as a minor. In other words, we give polynomial time algorithms that, for a given H-minor free graph G and positive integer k, output an H-minor free graph G′ on O(k) vertices such that G has a (connected) dominating set of size k if and only if G′ has. Prior to our work,...

A subset of vertices D of a graph G is a dominating set for G if every vertex of G not in D is adjacent to one in D. A dominating set for G is a connected dominating set if it induces a connected subgraph of G. The connected domination number of G, denoted by γc(G), is the minimum cardinality of a connected dominating set. Graph G is said to be k−γc−critical if γc(G) = k but γc(G+e) < k for eac...

Given a graphG, the k-dominating graph ofG, Dk(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 Dk(G) are adjacent if and only if the corresponding dominating sets of G differ by either adding or deleting a single vertex. The graph Dk(G) aids in studying the reconfiguration problem for dominating sets. In parti...

A problem open for many years is whether there is an FPT algorithm that given a graph G and parameter k, either: (1) determines that G has no k-Dominating Set, or (2) produces a dominating set of size at most g(k), where g(k) is some fixed function of k. Such an outcome is termed an FPT approximation algorithm. We describe some results that begin to provide some answers. We show that there is n...