Strong Roman Domination Number of Complementary Prism Graphs (original) (raw)

Roman Domination in Complementary Prism Graphs

A Roman domination function on a complementary prism graph GG c is a function f : V ∪ V c → {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number γR(GG c) of a graph G = (V, E) is the minimum of x∈V ∪V c f (x) over such functions, where the complementary prism GG c of G is graph obtained from disjoint union of G and its complement G c by adding edges of a perfect matching between corresponding vertices of G and G c. In this paper, we have investigated few properties of γR(GG c) and its relation with other parameters are obtained.

Partial Domination in Prisms of Graphs

2022

For any graph G = (V, E) and proportion p ∈ (0,1], a set S ⊆ V is a p-dominating set if |N[S]| |V | ≥ p. The p-domination number γp(G) equals the minimum cardinality of a p-dominating set in G. For a permutation π of the vertex set of G, the graph πG is obtained from two disjoint copies G1 and G2 of G by joining each v in G1 to π(v) in G2. i.e., V (πG) =V (G1)∪ V (G2) and E(G) = E(G1)∪E(G2)∪{{v,π(v)} : v ∈ V (G1),π(v) ∈ V (G2)}. The graph πG is called the prism of G with respect to π . In this paper, we find some relations between the domination and the p-domination numbers in the context of graph and its prism graph for particular values of p. Keywords/Phrases: Permutation graph, algebraic graph theory, prism graph

On the General Position Number of Complementary Prisms

Fundam. Informaticae, 2021

The general position number rmgp(G){\rm gp}(G)rmgp(G) of a graph GGG is the cardinality of a largest set of vertices SSS such that no element of SSS lies on a geodesic between two other elements of SSS. The complementary prism GoverlineGG\overline{G}GoverlineG of GGG is the graph formed from the disjoint union of GGG and its complement overlineG\overline{G}overlineG by adding the edges of a perfect matching between them. It is proved that rmgp(GoverlineG)len(G)+1{\rm gp}(G\overline{G})\le n(G) + 1rmgp(GoverlineG)len(G)+1 if GGG is connected and rmgp(GoverlineG)len(G){\rm gp}(G\overline{G})\le n(G)rmgp(GoverlineG)len(G) if GGG is disconnected. Graphs GGG for which rmgp(GoverlineG)=n(G)+1{\rm gp}(G\overline{G}) = n(G) + 1rmgp(GoverlineG)=n(G)+1 holds, provided that both GGG and overlineG\overline{G}overlineG are connected, are characterized. A sharp lower bound on rmgp(GoverlineG){\rm gp}(G\overline{G})rmgp(GoverlineG) is proved. If GGG is a connected bipartite graph or a split graph then rmgp(GoverlineG)inn(G),n(G)+1{\rm gp}(G\overline{G})\in \{n(G), n(G)+1\}rmgp(GoverlineG)inn(G),n(G)+1. Connected bipartite graphs and block graphs for which rmgp(GoverlineG)=n(G)+1{\rm gp}(G\overline{G})=n(G)+1rmgp(GoverlineG)=n(G)+1 holds are characterized. A family of block graphs is constructed in which the ${\rm g...

𝑘-Tuple Total Domination in Complementary Prisms

ISRN Discrete Mathematics, 2011

Letkbe a positive integer, and letGbe a graph with minimum degree at leastk. In their study (2010), Henning and Kazemi defined thek-tuple total domination numberγ×k,tGofGas the minimum cardinality of ak-tuple total dominating set ofG, which is a vertex set such that every vertex ofGis adjacent to at leastkvertices in it. IfG̅is the complement ofG, the complementary prismGG̅ofGis the graph formed from the disjoint union ofGandG̅by adding the edges of a perfect matching between the corresponding vertices ofGandG̅. In this paper, we extend some of the results of Haynes et al. (2009) for thek-tuple total domination number and also obtain some other new results. Also we find thek-tuple total domination number of the complementary prism of a cycle, a path, or a complete multipartite graph.

The Split Domination Number of a Prism Graph

Advances and Applications in Discrete Mathematics, 2015

A dominating set D of a graph () E V G , = is a split dominating set if the induced subgraph of D V − is disconnected. The split domination number () G s γ is the minimum cardinality of split dominating set of a graph G. In this paper, we establish some results on split domination in prism graphs. Particularly, we characterize the split domination in prism graph in terms of order and prism doublers of a dumble graph. Also, we characterize the domination and split domination in prism of corona graphs.

Open-independent, Open-locating-dominating Sets in Complementary Prism Graphs

Electronic Notes in Theoretical Computer Science, 2019

For a finite, simple, undirected graph G = (V (G), E(G)), an open-dominating set S ⊆ V (G) is such that every vertex in G has at least one neighbor in S. An open-independent, open-locating-dominating set S ⊆ V (G) (OLD OIN D-set for short) is such that no two vertices in G have the same set of neighbors in S and each vertex in S is open-dominated exactly once by S. The problem of deciding whether or not a given graph has an OLD OIN D-set is known to be N P-complete. The complementary prism of G is the graph GḠ, formed from the disjoint union of G and its complementḠ by adding the edges of a perfect matching between the corresponding vertices of G andḠ. We provided a logarithmic lower bound on the size of an OLD OIN D-set in any graph. Various properties of and bounds on OLD OIN D-sets in complementary prisms were presented and the cases of cliques, paths and cycles have been completely solved. It has been shown that for any graph with girth at least five, it can be decided in polynomial time whether or not its complementary prism has an OLD-OIND-set (and also the set can be found in polynomial time if it exists).

k-Independence on complementary prism graphs

Matemática Contemporânea, 2022

A subset S of vertices of a graph G is k-independent if each vertex in S has degree at most k − 1. The complementary prism of a graph G, denoted by GG, is a graph constructed by the disjoint union of G and its complement G by adding edges of a perfect matching between the corresponding vertices. We present some results on k-independent sets in complementary prisms, including sharp lower and upper bounds for the 2-independence number. Moreover, we present exact values for α 2 for the complementary prism of some particular graph classes.

On the convexity number for complementary prisms

ArXiv, 2018

In the geodetic convexity, a set of vertices SSS of a graph GGG is textitconvex\textit{convex}textitconvex if all vertices belonging to any shortest path between two vertices of SSS lie in SSS. The cardinality con(G)con(G)con(G) of a maximum proper convex set SSS of GGG is the textitconvexitynumber\textit{convexity number}textitconvexitynumber of GGG. The textitcomplementaryprism\textit{complementary prism}textitcomplementaryprism GoverlineGG\overline{G}GoverlineG of a graph GGG arises from the disjoint union of the graph GGG and overlineG\overline{G}overlineG by adding the edges of a perfect matching between the corresponding vertices of GGG and overlineG\overline{G}overlineG. In this work, we we prove that the decision problem related to the convexity number is NP-complete even restricted to complementary prisms, we determine con(GoverlineG)con(G\overline{G})con(GoverlineG) when GGG is disconnected or GGG is a cograph, and we present a lower bound when diam(G)neq3diam(G) \neq 3diam(G)neq3.

On the Geodetic Hull Number of Complementary Prisms

RAIRO Oper. Res., 2021

In the geodetic convexity, a set of vertices S of a graph G is convex if all vertices belonging to any shortest path between two vertices of S lie in S. The convex hull H(S) of S is the smallest convex set containing S. If H(S) = V (G), then S is a hull set. The cardinality h(G) of a minimum hull set of G is the hull number of G. The complementary prism GḠ of a graph G arises from the disjoint union of the graph G and Ḡ by adding the edges of a perfect matching between the corresponding vertices of G and Ḡ. A graph G is autoconnected if both G and Ḡ are connected. Motivated by previous work, we study the hull number for complementary prisms of autoconnected graphs. When G is a split graph, we present lower and upper bounds showing that the hull number is unlimited. In the other case, when G is a non-split graph, it is limited by 3.

k-Tuple Total Restrained Domination in Complementary Prisms

2015

In a graph G with δ(G) ≥ k ≥ 1, a k-tuple total restrained dominating set S is a subset of V(G) such that each vertex of V(G) is adjacent to at least k vertices of S and also each vertex ofV(G) − S is adjacent to at least k vertices of V(G) − S.Theminimumnumber of vertices of such sets inG is the k-tuple total restrained domination number ofG. In [k-tuple total restrained domination/domatic in graphs, BIMS], the author initiated the study of the k-tuple total restrained domination number in graphs. In this paper, we continue it in the complementary prism of a graph.