Vertex–Edge Roman {2}-Domination

Ahlam Almulhim () and Saiful Rahman Mondal
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Ahlam Almulhim: Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Ahsa 31982, Saudi Arabia
Saiful Rahman Mondal: Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Ahsa 31982, Saudi Arabia

Mathematics, 2025, vol. 13, issue 13, 1-13

Abstract: A vertex–edge Roman { 2 } -dominating function on a graph G = ( V , E ) is a function f : V ⟶ { 0 , 1 , 2 } satisfying that, for every edge u v ∈ E with f ( v ) = f ( u ) = 0 , ∑ w ∈ N ( v ) ∪ N ( u ) f ( w ) ≥ 2 . The weight of the function f is the sum ∑ a ∈ V f ( a ) . The vertex–edge Roman { 2 } -domination number of G , denoted by γ v e R 2 ( G ) , is the minimum weight of a vertex–edge Roman { 2 } -dominating function on G . In this work, we begin the study of vertex–edge Roman { 2 } -domination. We determine the exact vertex–edge Roman { 2 } -domination number for cycles and paths, and we provide a tight lower bound and a tight upper bound for the vertex–edge Roman { 2 } -domination number of trees. In addition, we prove that the decision problem associated with vertex–edge Roman { 2 } -domination is NP-complete for bipartite graphs.

Keywords: vertex–edge domination; Roman domination; Roman {2}-domination; paths; trees (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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