K_1,2-isolation In Graphs

The domination in graphs is one of the current research hotspots in the graph theory.This paper mainly studies the partial domination(the isolation)in graphs,and we determine a tight upper bound of the K1,2-isolation number for general graphs.A subset S?V(G)is called a F-isolating set of a graph G if G-N[S]contains no subgraph isomorphic to any F ∈ F,where F is a family of connected graphs.The F-isolation number of G,denoted by l(G,F),is the minimum cardinality of a Fisolating set in G.Denote that l(G,{K1,k+1})=l(G,K1,k+1)=lk(G).In particular,take k=1.Thus,any K1,2-isolating set D of G means G-N[D]consists of some isolated vertices and isolated edges only,and the Kl,2-isolation number l1(G)of G is the minimum cardinality of a Kl,2-isolating set D in G.This paper mainly proves that if G is a connected graph of order n and different from P3,C3 or C6,then l1(G)≤2/7n.Further,if girth(G)≥ 7,then this bound can be improved to n/4 unless G is a P3,a C7 or a C11.Both two upper bounds above are tight,and we characterize two infinite graph classes that can attain corresponding bounds.As an extension of Borg and Kaemawichanurat’s work on l1(G)for maximal outerplanar graphs in 2020,the focus of our research is to investigate the upper bound of l1(G)for general graphs.Moreover,the result under the girth condition generalize the result of Caro and Hansberg in trees in 2017.