The geodetic-dominating number of comb product graphs

A set of vertices S is called a geodetic-dominating set of G if every vertex outside S is adjacent to a vertex in S, and also is located inside a shortest path between two vertices in S. The geodeticdominating number of G is the minimum cardinality of geodetic-dominating sets of G. In this paper, we determine an exact value of the geodetic-dominating number of comb product graphs of any connected graphs of order at least two.


Introduction
In this paper, all graphs are assumed to be connected, finite, simple, and undirected. Let G be a graph. For a vertex z ∈ V (G), we recall that the open neighborhood and the closed neighborhood of z in G is defined as The domination number of G is the minimum cardinality of dominating sets of G. This concept provides several applications especially in protection strategies and business networking [10]. Interested readers are referred to a number of relevant literature mentioned in the references, including [16,24].
There are several modifications on domination concept in graph. Some of them are locatingdominating set [2,6,19,23], independent dominating set [4,14], Roman dominating set [9,13]. In this paper, we are interested to study another variant of domination in graph, namely geodeticdominating set.
A walk in G is a finite non-empty sequence W = v 0 e 1 v 1 e 2 v 2 ...e k v k where for 1 ≤ j ≤ k, v j is a vertex and for 1 ≤ i ≤ k, e i is an edge where v i−1 and v i are its end points. We can say that W is a v 0 − v k walk. A walk W is called a trail in case all edges of W are different. If all vertices of a trail W are also different, then W is called a path. The distance between vertices a, b ∈ V (G), denoted by d G (a, b), is the minimum number of edges of a − b paths in G. An a − b path with d G (a, b) edges is called an a − b geodesic. We denote I G [a, b] as the set of vertices which are located inside some a − b geodesics of G. For a non-empty set The set B then we called as a geodetic set of G in case I G [B] = V (G). The minimum cardinality of geodetic sets of G is called as the geodetic number of G, denoted by g(G). For references on geodetic number in graphs, see [3,5].
In this paper, let a set B ⊆ V (G) be both geodetic and dominating in G. The set B then we call as a geodetic-dominating set of G. The geodetic-dominating number of G, denoted by γ g (G), is the minimum cardinality of geodetic-dominating sets of G.
This topic was firstly introduced by Escuadro et al. [12]. They proved that for a connected graph G or order at least n ≥ 2, max{g(G), γ(G)} ≤ γ g (G) ≤ n. They also characterized all graphs of order n ≥ 2 with geodetic-dominating number 2, n, and n − 1. Some authors consider this topic to certain classes of graph. Hansberg and Volkmann [15] have shown that the geodeticdominating problem for chordal graphs is NP-complete. Meanwhile the geodetic-dominating number of tree graphs and triangle-free graphs, can be seen in [12]. Some other references on geodeticdominating number in graphs, see [7,8,18].
In this paper, we are interested to apply the geodetic-dominating concept to a product graphs. In this paper, we consider the comb product of connected graphs G and H. In chemistry [1], some classes of chemical graphs can be considered as the comb product graphs. The comb product of connected graphs G and H at vertex o ∈ V (H), denoted by G o H , is a graph obtained by taking one copy of G and |V (G)| copies of H and identifying the i-th copy of H at the vertex o to the i-th vertex of G. The vertex o ∈ V (H) then we call as the identifying vertex. This product graphs have been widely investigated in many areas, including metric distance problems [11,21,22] and graph labeling problems [17,20].
In this paper, we use some definitions in order to determine the geodetic-dominating number , we also use the notation G[S] which is a maximal subgraph of G induced by all vertices of S.

Geodetic-domination number of comb product graphs
In two lemmas below, we provide some properties of a dominating set and a geodetic set in two isomorphic graphs.
be an isomorphism between graphs A and B. The set S is a dominating set of A if and only if {θ(x)|x ∈ S} is a dominating set of B.
Note that x and y are adjacent in A if and only if θ(x) and θ(y) are adjacent in B. Therefore, If S dominates A, then we obtain If T dominates B, then we obtain is also contained in θ(x) − θ(y) path in B, and vice versa. So, z belongs to x − y geodesic if and only if θ(z) belongs to θ(x) − θ(y) geodesic. Therefore, If S is a geodetic set of A, then we obtain If T is a geodetic set of B, then we obtain Therefore, we obtain a direct consequences of Lemmas 2.1 and 2.2 in corollary below. Proof. The only vertex in V l which is adjacent to a vertex in  In some lemmas below, we consider some properties of the geodetic-dominating set of an induced subgraph of K o .
By Lemmas 2.5 and 2.7, we obtain a property of geodetic-dominating set of an induced subgraph of K o , which can be seen in corollary below.  We say that the graph H is of: Proof. For the identifying vertex o ∈ V (H), we recall the notation K o = G o H. We distinguish three cases.
If (S ∩ V l ) ∪ {(g l , o)} ∈ B l , then by considering Corollary 2.2, we have Otherwise, we have It follows that |S ∩ V l | ≥ γ g (H).
A contradiction.