Given a graph $$G=(V,E)$$ and an integer k, the Minimum Membership Dominating Set (MMDS) problem seeks to find dominating set $$S \subseteq V$$ of G such that for each $$v \in , $$\vert N[v] \cap S\vert $$ is at most k. We investigate parameterized complexity obtain following results MMDS problem. First, we show NP-hard even on planar bipartite graphs. Next, W[1]-hard parameter pathwidth (and t...

abstract. let $l$ be a lattice with the least element $0$. an element $xin l$ is a zero divisor if $xwedge y=0$ for some $yin l^*=lsetminus left{0right}$. the set of all zero divisors is denoted by $z(l)$. we associate a simple graph $gamma(l)$ to $l$ with vertex set $z(l)^*=z(l)setminus left{0right}$, the set of non-zero zero divisors of $l$ and distinct $x,yin z(l)^*$ are adjacent if and only...

Let G be a connected graph. A set W ⊆ V (G) is resolving hop dominating of if in and for every vertex v ∈ \ there exists u such that dG(u, v) = 2. S 1-movable S, either {v} or ((V S) ∩ NG(v)) (S {v}) ∪ {u} G. The domination number G, denoted by γ 1 mRh(G) the smallest cardinality This paper presents characterization sets join, corona lexicographic product graphs. Furthermore, this determines ex...