Isolation Number of Transition Graphs - Junhao Qu; Shumin Zhang (Mathematics)
Let <semantics>G=(V,E)</semantics> be a graph and <semantics>F</semantics> be a family of graphs; a subset (<semantics>S⊆V(G)</semantics>) is said to be an <semantics>F</semantics>-isolating set if <semantics>G[V(G)∖NG[S]]</semantics> does not contain F as a subgraph for all <semantics>F∈F</semantics>. The <semantics>F</semantics>-isolation number of G is the minimum cardinality of an <semantics>F</semantics>-isolating set (S) of G, denoted by <semantics>ι(G,F)</semantics>. When <semantics>F={K1,k+1}</semantics>, we use <semantics>ιk(G)</semantics> to define the <semantics>F</semantics>-isolation number (<semantics>ι(G,F)</semantics>). In particular, when <semantics>k=0</semantics>, we use the short form of <semantics>ι(G)</semantics> instead of <semantics>ι0(G)</semantics>. A subset (<semantics>S⊆V(G)</semantics>) is called an isolating set if <semantics>V(G)∖NG[S]</semantics> is an independent set of G. The isolation number of G is the minimum cardinality of an isolating set, denoted by <semantics>ι(G)</semantics>. In this paper, we mainly focus on research on the isolation number and <semantics>F</semantics>-isolation number of a <semantics>B(G)</semantics> graph, total graph and central graph of graph G.