2026· Far East Journal of Mathematical Sciences (FJMS)· Vol 143, pp. 4135-4157· 0 citations
TL;DR
Fundamental properties of the resilient domination number are established and corresponding bounds and explicit formulas for the resilient domination number are obtained.
Abstract
A resilient dominating set in a graph is a dominating set in which each vertex has at least one neighbor within the set and at least one neighbor outside the set, and the weakly induced subgraph of the set is connected. The smallest cardinality of such a set is the resilient domination number. This paper establishes fundamental properties of this parameter and determines its exact values for several graph families. Furthermore, resilient dominating sets are characterized for graphs constructed via the join, vertex corona, and edge corona operations, and corresponding bounds and explicit formulas for the resilient domination number are obtained.
A semitotal dominating set of a nontrivial connected graph G is a dominating set D of G such that every vertex in D is within distance two of another vertex in D. The semitotal domination number of G is the minimum cardinality among all semitotal dominating sets of G. In this article, we investigate the semitotal domin...
A. Cabrera-Martínez, J. L. López-Carmona, José M. Rodríguez et al.· Asian-European Journal of Ma...· 0 citations
For a subgraph of a vertex set of a graph to be strong efficient, all remaining vertices in the graph must have exactly one neighbor of equal or greater degree in the specified subgraph and all vertices in the specified subgraph are independent of each other. The strong efficient domination number is the cardinality of...
Betul Atay· Communications faculty of sc...· 0 citations
Let G be a nontrivial graph. A set D ⊆ V(G) is a double dominating set of G if |
\mathrm{N_G}
[v] ∩ D| ≥ 2 for every vertex v ∈ V(G), where
\mathrm{N_G}
[v] represents the closed neighborhood of v. The double domination number of G is the minimum cardinality among all double dominating sets of G. In this paper we...
A. Cabrera-Martínez, Ismael Rios-Villamar, J. Sigarreta· Filomat· 1 citation
Let G be a simple connected graph. We introduce certified weakly connected domination, where each vertex in a weakly connected dominating set S has either zero or at least two neighbors outside S. We establish fundamental relationships and determine the certified weakly connected domination number \gamma^{wc}_{cer}(G)...
Jamil J. Hamja, Faizal S. Hajal, Mercedita A. Langamin et al.· International Journal of Mat...· 0 citations
Let $G$ be a graph with vertex set $V(G)$. A set $I\subseteq V(G)$ is an independent dominating set of $G$ if no two vertices in $I$ are adjacent and every vertex in $V(G)\setminus I$ is adjacent to at least one vertex in $I$. The independent domination number of $G$ is the minimum cardinality among all independent dom...
A. Cabrera-Martínez, J. L. López-Carmona, Ismael Rios-Villamar et al.· 1 citation
Let G be a non-trivial graph. A dominating set I ⊂ V (G) is called an Isolate Pendant Dominating set (IPD-set) if the induced subgraph 〈I〉 has maximum degree at most one and contains at least one isolate vertex and at least one pendant vertex. The minimum cardinality of an IPD-set is called the IP domination number a...
AL. Yakavi, A. Mydeen Bibi· Reserche operationelle· 0 citations
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