A class of graphs is said to be biclique-free if there is an integer t such that no graph in the class contains Kt,t as a subgraph. Large families of graph classes, such as any nowhere dense class of graphs or d-degenerate graphs, are biclique-free. We show that various domination problems are fixed-parameter tractable on biclique-free classes of graphs, when parameterizing by both solution siz...

Many NP-hard problems, such as Dominating Set, are FPT parameterized by clique-width. For graphs of clique-width k given with a kexpression, Dominating Set can be solved in 4knO(1) time. However, no FPT algorithm is known for computing an optimal k-expression. For a graph of clique-width k, if we rely on known algorithms to compute a (23k − 1)expression via rank-width and then solving Dominatin...

Many NP-hard problems, such as Dominating Set, are FPT parameterized by clique-width. For graphs of clique-width k given with a kexpression, Dominating Set can be solved in 4knO(1) time. However, no FPT algorithm is known for computing an optimal k-expression. For a graph of clique-width k, if we rely on known algorithms to compute a (23k − 1)expression via rank-width and then solving Dominatin...

Roman dominating function} on a digraph $D$ with vertex set $V(D)$ is a labeling$fcolon V(D)to {0, 1, 2}$such that every vertex with label $0$ has an in-neighbor with label $2$. A set ${f_1,f_2,ldots,f_d}$ ofRoman dominating functions on $D$ with the property that $sum_{i=1}^d f_i(v)le 2$ for each $vin V(D)$,is called a {em Roman dominating family} (of functions) on $D$....

A graph is k-connected if it has k internally-disjoint paths between every pair of nodes. A subset S of nodes in a graph G is a kconnected set if the subgraph G[S] induced by S is k-connected; S is an m-dominating set if every v ∈ V \ S has at least m neighbors in S. If S is both k-connected and m-dominating then S is a k-connected m-dominating set, or (k,m)-cds for short. In the k-Connected mD...

A k-dominating set of a graph G is a subset D of the vertices of G such that every vertex of G is either in D or at distance at most k from a vertex in D. It is of interest to find k-dominating sets of small cardinality. In this paper we consider simple randomised greedy algorithms for finding small k-dominating sets of regular graphs. We analyse the average-case performance of the most efficie...

let $k$ be a positive integer. a subset $s$ of $v(g)$ in a graph $g$ is a $k$-tuple total dominating set of $g$ if every vertex of $g$ has at least $k$ neighbors in $s$. the $k$-tuple total domination number $gamma _{times k,t}(g)$ of $g$ is the minimum cardinality of a $k$-tuple total dominating set of $g$. if$v(g)=v^{0}={v_{1}^{0},v_{2}^{0},ldots ,v_{n}^{0}}$ and $e(g)=e_{0}$, then for any in...

Let G = (V, E) be a graph without isolated vertices. A set D ⊆ V is a d-distance paired-dominating set of G if D is a d-distance dominating set of G and the induced subgraph 〈D〉 has a perfect matching. The minimum cardinality of a d-distance paired-dominating set for graph G is the d-distance paired-domination number, denoted by γd p(G). In this paper, we study the ddistance paired-domination n...

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