For each k, k 1 , k 2 , k 3 , k 4 ∈ N, we will denote by P k (k 1 , k 2), (k 3 , k 4) a tree of k + k 1 + k 2 + k 3 + k 4 + 1 vertices with the degree sequence (1, 1, 1, 1, 2, 2, 2,. .. , 2, 3, 3). Let α k 1 , β k 2 , σ k 3 , and δ k 4 be all four endpoints of the graph. Let the distance between both vertices of degree 3 be equal to k. A subset S of vertices of a graph P k (k 1 , k 2), (k 3 , k 4) is a dominating set of P k (k 1 , k 2), (k 3 , k 4) if every vertex in V P k (k 1 , k 2), (k 3 , k 4) − S is adjacent to some vertex in S. We investigate the dominating set of minimum cardinality of a graph P k (k 1 , k 2), (k 3 , k 4) to obtain the domination number of this graph. Finally, we determine an upper bound on the domination number of a graph P k (k 1 , k 2), (k 3 , k 4) .

Let G 1 and G 2 be two vertex-disjoint graphs with H 1 a subgraph of G 1 and H 2 a subgraph of G 2 . Let f : H 1 → H 2 be an isomorphism between these subgraphs. The glued graph of G 1 and G 2 at H 1 and H 2 with respect to f is the graph that results from combining G 1 ∪ G 2 by identifying the subgraphs H 1 and H 2 according to the isomorphism f between H 1 and H 2 . We refer G 1 and G 2 as its original graphs and refer H as its clone where H is a copy of H 1 and H 2 . In this paper, we enumerate all non-isomorphic resulting glued graphs between two paths at connected clones. Moreover, we also give the characterization of the glued graph at a connected clone.