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A vertex of a graph is locally connected if its neighborhood is connected. A graph G is locally connected if every vertex of G is locally connected. A graph is called claw-free if it does not contain a copy of K 1,3 as an induced subgraph. In this paper, we provide a constructive characterization of 4-connected 4-regular locally connected claw-free graphs. From its proof, we can give a linear polynomial-time algorithm to construct a 4-connected 4-regular locally connected claw-free graph.

Linear Polynomial-Time Algorithms To Construct 4-Connected 4-Regular Locally Connected Claw-Free Graphs