Nowhere-zero 3-flows of claw-free graphs

Let AA denote an abelian group and GG be a graph. If a graph G∗G∗ is obtained by repeatedly contracting nontrivial AA-connected subgraphs of GG until no such a subgraph left, we say GG can be AA-reduced to G∗G∗. A graph is claw-free if it has no induced subgraph K1,3K1,3. Let N1,1,0N1,1,0 denote the graph obtained from a triangle by adding two edges at two distinct vertices of the triangle, respectively. In this paper, we prove that if GG is a simple 2-connected {claw,N1,1,0}{claw,N1,1,0}-free graph, then GG does not admit nowhere-zero 3-flow if and only if GG can be Z3Z3-reduced to two families of well characterized graphs or GG is one of the five specified graphs.