Note that if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] contains in particular all edges of the

induced subgraph G[X] of G on X.

A dominating set S of a graph G without isolated vertices is called the neighborhood total dominating set (ntd-set) if the

induced subgraph N(S) has no isolated vertices.

For a subgraph H of a graph G, G-H denotes the

induced subgraph by V(G) - V(H), and G[S] denotes the

induced subgraph by S for S [subset or equal to] V(G).

A graph is said to be a subgraph of if and If ' contains all edges of that join two vertices in then is said to be the subgraph induced or spanned by , and is denoted by Thus, a subgraph of is an

induced subgraph if If , then is said to be a spanning subgraph of Two graphs are isomorphic if there is a correspondence between their vertex sets that preserves adjacency.

Acharya, Germina, and Ajitha [1] have shown that every graph can be embedded as an

induced subgraph of a strongly multiplicative graph.

A graph is called claw-free if it has no

induced subgraph isomorphic to [K.

Let us draw the

induced subgraph of the graph G1 (of the graph G2 respectively) containing the vertex 6 (the vertex Me respectively) and its neighbors (see Figure 18).

A dominating set D of a graph G is said to be split and non-split dominating set if the

induced subgraph <V\D> disconnected and <V\D> connected respectively.

If the

induced subgraph of is connected, then is a connected dominating set (CDS).

An intersection graph of a set of 2 x 2 dense blocks is an

induced subgraph of the so called X-grid which consists of the usual 2 dimensional grid, and diagonals for each grid square.

The

induced subgraph on the neighbors of i is vector (k - 1)-colorable.

For every graph H there is a constant [sigma] > 0 such that every graph on n vertices which does not contain H as an

induced subgraph has a clique or independent set of size at least [n.