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 adjacen...

Let G be a graph. A set S of vertices in G dominates the graph if every vertex of G is either in S or a neighbor of a vertex in S. Finding a minimal cardinality set which dominates the graph is an NP-complete problem. The graph G is well-dominated if all its minimal dominating sets are of the same cardinality. The complexity status of recognizing well-dominated graphs is not known. We show that...

A dominating set D in a graph is a subset of its vertex set such that each vertex is either in D or has a neighbour in D. In this paper, we are interested in an output-sensitive enumeration algorithm of (inclusionwise) minimal dominating sets in graphs, called Dom problem. It was known that this problem can be polynomially reduced to the well known Transversal problem in hypergraphs. We show th...

Let G = (V,E) be a connected undirected graph. For any vertex v ∈ V , the closed neighborhood of v is N [v] = {v} ∪ {u ∈ V | uv ∈ E }. For S ⊆ V , the closed neighborhood of S is N [S] = ⋃ v∈S N [v]. The subgraph weakly induced by S is 〈S〉w = (N [S], E ∩ (S × N [S])). A set S is a weakly-connected dominating set of G if S is dominating and 〈S〉w is connected. The weakly-connected domination numb...

Signed link prediction in social networks aims to reveal the underlying relationships (i.e., links) among users nodes) given their existing positive and negative interactions observed. Most of prior efforts are devoted learning node embeddings with graph neural (GNNs), which preserve signed network topology by message-passing along edges facilitate downstream task. Nevertheless, graph-based app...

The cardinality of a maximum minimal dominating set of a graph is called its upper domination number. The problem of computing this number is generally NP-hard but can be solved in polynomial time in some restricted graph classes. In this work, we consider the complexity and approximability of the weighted version of the problem in two special graph classes: planar bipartite, split. We also pro...

Let G be a graph with integral edge weights. A function d : V (G) → Zp is called a nowhere 0 mod p domination function if each v ∈ V satisfies ( d(v) + ∑ u∈N(v) w(u, v)d(u) ) 6= 0 mod p, where w(u, v) denotes the weight of the edge (u, v) and N(v) is the neighborhood of v. The subset of vertices with d(v) 6= 0 is called a nowhere 0 mod p dominating set. It is known that every graph has a nowher...

Domination in graphs’ has been studied extensively and at present it is an emerging area of research in graph theory. An introduction and an extensive overview on domination in graphs and related topics is surveyed and detailed in the two books by Haynes et al. [1,2]. Product of graphs occurs naturally in discrete mathematics as tools in combinatorial constructions. They give rise to an importa...

A pair of vertices of a graph is called a dominating pair if the vertex set of every path between these two vertices is a dominating set of the graph. A graph is a weak dominating pair graph if it has a dominating pair. Further, a graph is called a dominating pair graph if each of its connected induced subgraphs is a weak dominating pair graph. Dominating pair graphs form a class of graphs cont...