SECURE DOMINATION IN LICT GRAPHS

For any graph G = (V,E), lict graph η(G) of a graph G is the graph whose vertex set is the union of the set of edges and the set of cut-vertices of G in which two vertices are adjacent if and only if the corresponding edges are adjacent or the corresponding members of G are incident. A secure lict dominating set of a graph η(G) , is a dominating set F ⊆ V (η(G)) with the property that for each v1 ∈ (V (η(G)) − F ), there exists v2 ∈ F adjacent to v1 such that (F −{v2})∪{v1} is a dominating set of η(G). The secure lict dominating number γse(η(G)) of G is a minimum cardinality of a secure lict dominating set of G. In this paper, many bounds on γse(η(G)) are obtained and its exact values for some standard graphs are found in terms of parameters of G. Also its relationship with other domination parameters is investigated. Mathematics Subject Classification: 05C69.


Introduction
The graphs considered here are finite, connected, undirected without loops or multiple edges and without isolated vertices. As usual n and q denote the number of vertices and edges of a graph G. For any undefined term or notation in this paper can be found in Harary [1]. A set D ⊆ V is a dominating set of G if every vertex in V −D is adjacent to some vertex in D. The dominating number γ(G) of G is the minimum cardinality of a dominating set D. A secure dominating set of G is a dominating set D ⊆ V (G) with the property that for each u ∈ V (G) − D, there exists v ∈ D adjacent to u such that (D − {v}) − {u} is a dominating set.
The total domination of lict graph has been studied by [2]. The lict graph η(G) of a graph G is the graph whose vertex set is the union of the set of edges and the set of cut-vertices of G in which two vertices are adjacent if and only if the corresponding edges are adjacent or the corresponding members of G are incident. The secure domination has been intensively studied by [3,4]. The secure lict dominating set of a graph η(G) , is a dominating set F ⊆ V (η(G)) with the property that for each v 1 ∈ (V (η(G)) − F ), there exists v 2 ∈ F adjacent to v 1 such that (F − {v 2 }) ∪ {v 1 } is a dominating set of η(G). The secure lict dominating number γ se (η(G)) of G is a minimum cardinality of secure lict dominating set of graph G. For complete review on the topic of domination [5]. The vertex independence number β 0 (G) is the maximum cardinality among the independent set of vertices of G. L(G) is the line graph of G, γ e (G) is edge domination number, γ ′ s (G) is the secure edge dominating number, γ t (G) is the total dominating number , γ ns (G) is the non-split dominating number and χ(G)is the chromatic number of G. The degree of a edge [6] is the number of lines adjacent to it. The minimum (maximum) degree of an edge in G is denoted by δ ′ (∆ ′ ). A subdivision of an edge e = uv of a graph G is the replacement of an edge e by a path (u, v, w) where w / ∈ E(G). The graph obtained from G by subdividing each edge of G exactly once is called the subdivision graph of G and is denoted by S(G). For any real number X,⌈X⌉ denotes the smallest integer not less than X and ⌊X⌋ denotes the greatest integer not greater than X. In this paper we established the relationship of this concept with the other domination parameters is investigated.

Main results
Theorem 2.1. First we list out the exact values of γ se (η(G)) for some standard graphs: (i) For any cycle C n with n ≥ 3 vertices, (ii) For any path P n with n ≥ 4 vertices,γ se (η(P n )) = n − 2.
Case 1: If G is a connected graph with n = 3, then G is either K 1,2 or C 3 , by using Theorem 2.1(i) and Theorem 2.1(iii), γ se (η(G)) = 1. Case 2: If G is a connected graph with n ≥ 4. Let D = {e} be the secure dominating set of γ se (η(G)). To prove that G = K 1,n−1 , we assume contrary that G ̸ = K 1,n−1 . We consider the following subcases: Subcase 1: Let F = K 1,n−1 and let the endvertices v 1 , v 2 ∈ V (F ), such that the graph G is obtained form F by adding the edge e 1 = (v 1 , v 2 ) / ∈ E(F ). It follows that the set (D − {e}) ∪ {e 1 } is not a dominating set of η(G). This implies that D is not a dominating set of η(G), which is a contradiction. Thus G = K 1,n−1 . Subcase 2: Let F = K 1,n−1 and an endvertex v 1 ∈ V (F ), such that the graph G is obtained form F by adding the vertex v ∈ V (F )and the edge e 1 = (v, v 1 ). It follows that the set (D − {e}) ∪ {e 1 } is not an secure dominating set of η(G). This implies that D is not a dominating set of η(G), which is a contradiction. Thus G = K 1,n−1 .

Theorem 2.3. For any graph
Proof. Let D be a secure edge dominating set of G and let B be the corresponding vertices of D in η(G). We consider the following cases:  Therefore γ se (η(G)) = γ e (G). Otherwise the corresponding vertices of {D ∪ e i ∪ c i } in η(G) is the secure dominating set of η(G). Therefore γ se (η(G)) ≥ γ e (G).
For Equality: The result follows from Theorem 2.1(ii) and Theorem 2.1(iii).
Proof. Let A be the set of cut-vertices of G with |A| = m and let

Corollary 2.8. If every vertex of G is adjacent to an end vertex, then
Proof. Since every vertex of G is adjacent to an end vertex then, γ(G) = γ t (G) = γ ns (G) = m and by using Theorem 2.7, the result follows.
and ∆ ′ is the maximum degree of an edge.
Proof. Let e be an edge with degree ∆ ′ and let S be the set of edges adjacent to e in G. Then E(G) − S is the lict dominating set of G. We consider the following cases: Case 1: If G is a non-separable graph, then for every edge e 1 ∈ S, there exist an edge e 2 ∈ E(G) − S, e 1 ∈ N (e 2 ) such that the corresponding vertices of Case 2: If G is a separable graph. We consider the following subcases: where v 1 or v 2 or both, are the cut-vertices of graph G. Therefore by using Theorem 2.7 , is not a null graph and The result follows from Case (1) and Case (2). We consider the following cases.
there exists an edge e j which is incident with v i and e j ∈ A such that the corresponding vertices of Theorem 2.11. For any connected graph G, γ se (η(G)) ≤ n − 2, n ≥ 3.
We consider the following cases.
We consider the following subcases: (i) If |F | ̸ = ϕ and if v j is a not a cut-vertex or e j ∈ N (e i ) ∈ D, then γ se (η(G)) = γ ′ s (G). Otherwise there exists atleast one vertex in η(G) which is not covered by C. Therefore A ∈ γ se set of η(G).
Proof. Let D be the maximum vertex independence set of T and let A be the γ se set of η(G). Since every vertex is adjacent to an end vertex, then A will contains all the vertices adjacent to endvertices of T with |A| ≤ β 0 (T ). Therefore by using Theorem 2.7, we have γ se (η(T )) ≤ β 0 (T ).