2025/05/13 by Shuai Kou, Kou, Shuai, Chengfu Qin +5
Computer Science · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2505.08185
We call a pair of non-adjacent vertices in G a non-edge. Contraction of a non-edge u, v in G is the replacement of u and v with a single vertex z and then making all the vertices that are adjacent to u or v adjacent to z. A non-edge u, v is said to be contractible in a k-connected graph G, if the resulting graph after its contraction remains k-connected. Tsz Lung Chan characterized all 3-connected graphs (finite or infinite) that does not contain any contractible non-edges in 2019, and posed the problem of characterizing all 3-connected graphs that contain exactly one contractible non-edge. In this paper, we solve this problem.