On Some Special Induced-cycle-nice Graphs

In a graph G,an even cycle C is called a nice cycle of G,if G-V(C)has a perfect matching.A graph G is an induced-cycle-nice graph,if any induced even cycle of G is a nice cycle.A Halin graph is a graph H:=T U C,where T is a plane tree on at least four vertices in which no vertex has degree two,and C is a cycle connecting the leaves of T in the cyclic order determined by the embedding of T.A caterpillar tree is a tree which leaves a path after removing vertices of degree one.When the tree T1 is a caterpillar tree,the Halin graph is called Caterpillar Halin graph.For induced-cycle-nice graphs,there are some good results,including the complete characterization of induced-cycle-nice 2-edge-connected claw-free cubic graphs and the degree conditions of a graph to be an induced-cycle-nice graph.In this paper,we study some special induced-cycle-nice graphs.The main results are the followings:·Some complete characterizations of special induced-cycle-nice graphs,in-cluding the Cartesian product of path and path,the Cartesian of path and cycle,the power of paths and cycles.·Complete characterization of induced-cycle-nice graph of connected claw-free cubic graphs with cut edges.·Partial characterizations of induced-cycle-nice Caterpillar Halin Graphs.