Given a graph model G=(V, E) and a vertex subset S [??] V, [G.sub.S]=(S,E(S)) is a

induced subgraph of G for S.

The graph G is signed-eliminable with signedelimination ordering v : V(G) [right arrow] {0, ..., l} if v is bijective and, for every three vertices [[upsilon].sub.i],[[upsilon].sub.j],[[upsilon].sub.k] [member of] V(G) with v([[upsilon].sub.i]), v([[upsilon].sub.j]) < v([[upsilon].sub.k]), the

induced subgraph [mathematical expression not reproducible] satisfies:

Given a nonempty vertex subset V' of V, the

induced subgraph by V' in G, denoted by G[V'], is a graph, whose vertex set is V' and the edge set is the set of all the edges of G with both endpoints in V'.

Then bicyclic networks can be partitioned into two classes: the class of graphs which contain [infinity]-graph as its

induced subgraph and the class of graphs which contain [theta]-graph as its

induced subgraph.

A strongly connected component of G(V, E, A) is an

induced subgraph and is subjected to be strongly connected.

A parallel closure of a graph is an

induced subgraph on two vertices.

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).

We can then define the degree-limited subgraph [G.sub.i] = G[[V.sub.i]] as the

induced subgraph created from vertices of G which have degree less than or equal to i.

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.

A subgraph H is called an

induced subgraph of X if for any a,b [member of] V(H), {a, b} [member of] E(H) if and only if {a, b} [member of] E(X).