lexicographic product graphs
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Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1282
Author(s):  
Ana Almerich-Chulia ◽  
Abel Cabrera Martínez ◽  
Frank Angel Hernández Mira ◽  
Pedro Martin-Concepcion

Let G be a graph with no isolated vertex and let N(v) be the open neighbourhood of v∈V(G). Let f:V(G)→{0,1,2} be a function and Vi={v∈V(G):f(v)=i} for every i∈{0,1,2}. We say that f is a strongly total Roman dominating function on G if the subgraph induced by V1∪V2 has no isolated vertex and N(v)∩V2≠∅ for every v∈V(G)\V2. The strongly total Roman domination number of G, denoted by γtRs(G), is defined as the minimum weight ω(f)=∑x∈V(G)f(x) among all strongly total Roman dominating functions f on G. This paper is devoted to the study of the strongly total Roman domination number of a graph and it is a contribution to the Special Issue “Theoretical Computer Science and Discrete Mathematics” of Symmetry. In particular, we show that the theory of strongly total Roman domination is an appropriate framework for investigating the total Roman domination number of lexicographic product graphs. We also obtain tight bounds on this parameter and provide closed formulas for some product graphs. Finally and as a consequence of the study, we prove that the problem of computing γtRs(G) is NP-hard.







Author(s):  
Abel Cabrera Martinez ◽  
Alejandro Estrada-Moreno ◽  
Juan Alberto Rodriguez-Velazquez




2021 ◽  
Vol 40 (2) ◽  
pp. 385-398
Author(s):  
Elias John Thomas ◽  
Ullas Chandran S. V.

An independent set S of vertices in a graph G is an independent position set if no three vertices of S lie on a common geodesic. An independent position set of maximum size is an ip-set of G. The cardinality of an ip-set is the independent position number, denoted by ip(G). In this paper, we introduce and study the independent position number of a graph. Certain general properties of these concepts are discussed. Graphs of order n having the independent position number 1 or n − 1 are characterized. Bounds for the independent position number of Cartesian and Lexicographic product graphs are determined and the exact value for Corona product graphs are obtained. Finally, some realization results are proved to show that there is no general relationship between independent position sets and other related graph invariants.



2020 ◽  
Vol 284 ◽  
pp. 290-300 ◽  
Author(s):  
Abel Cabrera Martínez ◽  
Suitberto Cabrera García ◽  
J.A. Rodríguez-Velázquez


2020 ◽  
Vol 282 ◽  
pp. 152-161
Author(s):  
Tianlong Ma ◽  
Jinling Wang ◽  
Mingzu Zhang ◽  
Xiaodong Liang


Author(s):  
Abel Cabrera Martínez ◽  
Juan Alberto Rodríguez-Velázquez


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