Geodesic
Deficiency and Forbidden Subgraphs of Connected, Locally-Connected Graphs
Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 1, pp. 195-208

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A graph G is locally-connected if the neighbourhood N_G (v) induces a connected subgraph for each vertex v in G. For a graph G, the deficiency of G is the number of vertices unsaturated by a maximum matching, denoted by def (G). In fact, the deficiency of a graph measures how far a maximum matching is from being perfect matching. Saito and Xiong have studied subgraphs, the absence of which forces a connected and locally-connected graph G of sufficiently large order to satisfy def (G) ≤ 1. In this paper, we extend this result to the condition of def (G) ≤ k, where k is a positive integer. Let β_0 = 1/2 (3+√(8k+17) ) −1, we show that K_1,2, K_1,3, . . ., K_1,β_0, K_3 or K_2 2K_1 is the required forbidden subgraph. Furthermore, we obtain some similar results about 3-connected, locally-connected graphs. Key Words: deficiency, locally-connected graph, matching, forbidden subgraph.
Keywords: 05C40, 05C70 
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     author = {Li, Xihe and Wang, Ligong},
     title = {Deficiency and {Forbidden} {Subgraphs} of {Connected,} {Locally-Connected} {Graphs}},
     journal = {Discussiones Mathematicae. Graph Theory},
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Li, Xihe; Wang, Ligong. Deficiency and Forbidden Subgraphs of Connected, Locally-Connected Graphs. Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 1, pp. 195-208. http://geodesic.mathdoc.fr/item/DMGT_2020_40_1_a12/