A distance-k dominating set S of a directed graph D is a set of vertices such that for every vertex v of D, there is a vertex u ∈ S at distance at most k from it. Minimum distance-k dominating set is especially important in communication networks for distributed data structure and for server placement. In this paper, we shall present simple distributed algorithms for finding the unique minimum ...

We study dominating sets whose induced subgraphs have a bounded diameter. Such sets were used in recent papers by Kim et al. and Yu et al. to model virtual backbones in wireless networks where the number of hops required to forward messages within the backbone is minimized. We prove that for any fixed k ≥ 1 it is NP-complete to decide whether a given graph admits a dominating set whose induced ...

Let K be an imaginary quadratic field where p splits. We study signed Selmer groups for non-ordinary modular forms over the anticyclotomic ℤ -extension of K, showing that one inclusion Iwasawa main conjecture involving p-adic L-function Bertolini–Darmon–Prasanna implies their μ-invariants vanish. This gives alternative method to reprove a recent result Matar on vanishing plus and minus elliptic...

We show that the problem k-DOMINATING SET and its several variants including k-CONNECTED DOMINATING SET, k-INDEPENDENT DOMINATING SET, and k-DOMINATING CLIQUE, when parameterized by the solution size k, are W[1]-hard in either multiple-interval graphs or their complements or both. On the other hand, we show that these problems belong to W[1] when restricted to multipleinterval graphs and their ...

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...

For any non-square integer multiplier \(k\), there is an infinity of triangular numbers multiple other numbers. We analyze the congruence properties indices \(\xi\) Remainders in relations modulo \(k\) come always pairs whose sum equal \((k-1)\), include 0 and only \((k-1)\) if prime, or odd power a even square plus one minus two. If twice number \(n\), set remainders includes also \(n\) \((n^{...

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 $D$....