Analysis of Graphs with Signed Domination Number: Theoretical Insights and Applications in Network Optimization

Abstract The analysis of graphs with signed domination numbers, a concept defined by the signed dominating function f:V→{−1,1}f: V \rightarrow \{-1, 1\}f:V→{−1,1} on a graph G=(V,E)G = (V, E)G=(V,E), where ∑u∈N[v]f(u)≥1\sum_{u \in N[v]} f(u) \geq 1∑u∈N[v]f(u)≥1 for each vertex v∈Vv \in Vv∈V and the minimum weight ∑v∈Vf(v)\sum_{v \in V} f(v)∑v∈Vf(v) denoted as γS(G)\gamma_S(G)γS(G), extends the classical domination number by incorporating negative assignments to provide a nuanced perspective on vertex influence within a graph, and has become a significant area of interest due to its wide-ranging applications in network optimization such as efficient monitoring in communication networks using minimal devices, enhancing resource allocation strategies, and offering dual insights into supportive and antagonistic interactions in social network analysis, while recent research has explored its variability across graph classes like Cayley graphs, regular graphs, and trees, established bounds on parameters for various graph families, and introduced extensions such as the signed Roman domination number, revealing intricate relationships between graph structures and domination characteristics, ultimately contributing to a richer theoretical framework with practical implications for designing algorithms and methodologies in network design, resource allocation, and system monitoring across diverse fields.