Captive domination in graphs

Abstract

In this paper, a new definition of graph domination called “Captive Domination” is introduced. The proper subset of the vertices of a graph [Formula: see text] is a captive dominating set if it is a total dominating set and each vertex in this set dominates at least one vertex which doesn’t belong to the dominating set. The inverse captive domination is also introduced. The lower and upper bounds for the number of edges of the graph are presented by using the captive domination number. Moreover, the lower and upper bounds for the captive domination number are found by using the number of vertices. The condition when the total domination and captive domination number are equal to two is discussed and obtained results. The captive domination in complement graphs is discussed. Finally, the captive dominating set of paths and cycles are determined.

Examining Captive and Inverse Captive Domination in Selected Graphs and Their Complements

“…Definition 2.1 [1] Let G=(V, E) be a graph without isolated vertices, a subset D ⊂ V(G) is a captive dominating set (CDS) in G if G[D] has no isolated vertex (D is a total dominating set), and each v in set D is adjacent to at least one vertex in V-D. The minimum cardinality of a captive dominating set (CDN) of G denoted by γ ca (G), is called captive domination number.…”

“…Let 𝑁(𝑣) = {𝑢 ∈ 𝑉, 𝑢𝑣 ∈ 𝐸} be the open neighborhood of a vertex v and 𝑁[𝑣] = 𝑁(𝑣) ∪ {𝑣} is the closed neighborhood set. Also, let G[D] be the subgraph of 𝐺 induced by vertices in D [1]. A subset 𝐷 ⊆ 𝑉 (𝐺) is a dominating set of G if 𝑁(𝑣) ∩ 𝐷 ≠ ∅; for all 𝑣 ∈ 𝑉 − 𝐷.…”

“…The reader can be found all notions not mentioned [19][20][21][22]. The captive domination is initiated by Al-Harere et al [1], and they get many properties and bounded. the main targeted applications are placing security cameras, or personnel on a site, minimizing the effect of losing a dominating vertex by keeping another vertex with the same capabilities and each dominating vertex is dominating other vertices outside the dominating setin addition to dominating a neighbouring vertex in the dominating set.…”

Calculating Modern Roman Domination of Fan Graph and Double Fan Graph

“…In mathematics, this concept appeared in many fields including fuzzy graph [4][5][6] topological graph [7], polynomials domination [8,9], and others [10,11]. Additionally, many new definitions in this concept have been used, depending on putting some conditions on the dominating set [12], [13][14][15][16], out of dominating set [17][18][19] or on the two together as in [13]. A Roman dominating function on a graph 𝐺 = (𝑉; 𝐸) is a function 𝑓: 𝑉(𝐺) → {0,1,2}satisfying the condition that every vertex 𝑢 for which 𝑓(𝑢) = 0 is adjacent to at least one vertex 𝑣 for which 𝑓(𝑣) = 2, [20,21].…”

Domination in Fuzzy Directed Graphs

“…Extended forms of domination in graphs have been vast in the domain literature. Some very recent forms include broadcast domination [18], pitchfork domination [19], Roman domination [20], double Roman domination [21], triple Roman domination [22], captive domination [23], outer-convex domination [24], and paired domination [25], among others. The trajectory of these topics has been exponential in the last decade.…”