The structure of 2-matching connected graphs

Constructive characterizations of classical connectivity have been studied by many researchers.Whitney states that a graph is 2-connected if and only if it is a cycle or can be obtained from a cycle by repeatedly adding ears.Tutte asserts that a graph is 3-connected if and only if it is a wheel or can be obtained from a wheel by repeatedly adding edges and splitting vertices of degree more than three.Slater determines that the class of 4connected graphs is the class of graphs obtained from K 5 by finite sequences of edge addition, 4-soldering, 4-vertex-splitting, 4-edge-splitting, and 3-fold-4-vertex-splitting. For an integer k,is the set of vertices covered by M .In this paper, we present that a graph G is 2-matching connected if and only if either G ∈ {C 4 , K 4 } or it can be obtained from a C 4 by repeatedly applying the following three operations.(i) Adding a new edge uv for a pair of nonadjacent vertices u and v where {u, v} is not a vertex cut of G.(ii) Adding an ear uwv, where w is a new vertex and u, v are a pair of nonadjacent vertices of G.H. Li, M. Ma, S. Zhao and J. Liu(iii) Replacing a vertex z by two new adjacent vertices z 1 and z 2 , and making z i adjacent to every vertex of N i such that d(z i ) ≥ 2 for i ∈ {1, 2}, where N 1 ∪ N 2 = N (z) and N 1 ∩ N 2 = ∅.