Antoshyna, KaterynaKozerenko, Sergiy2025-03-122025-03-122025Antoshyna K. Graphs with odd and even distances between non-cut vertices / Kateryna Antoshyna, Sergiy Kozerenko // Opuscula Mathematica. - 2025. - Vol. 45, Issue 1. - P. 5-25. - https://doi.org/10.7494/OpMath.2025.45.1.51232−92742300−6919https://doi.org/10.7494/OpMath.2025.45.1.5https://ekmair.ukma.edu.ua/handle/123456789/33918We prove that in a connected graph, the distances between non-cut vertices are odd if and only if it is the line graph of a strong unique independence tree. We then show that any such tree can be inductively constructed from stars using a simple operation. Further, we study the connected graphs in which the distances between non-cut vertices are even (shortly, NCE-graphs). Our main results on NCE-graphs are the following: we give a criterion of NCE-graphs, show that any bipartite graph is an induced subgraph of an NCE-graph, characterize NCE-graphs with exactly two leaves, characterize graphs that can be subdivided to NCE-graphs, and provide a characterization for NCE-graphs which are maximal with respect to the edge addition operation.ennon-cut vertexgraph distanceline graphblockstrong unique independence treearticleArticle