On local properties of finite graphs

On extremal sizes of locally k-tree graphs

“…The graphs with local property P for |P| = 1 have been studied by many authors. The study has been inspired by the Trahtenbrot-Zykov problem [23] whether, given a graph H, there exists a graph G which is locally constant, namely locally H. Summaries of the results of this type can be found in the survey papers by Hell [10] and Sedláček [19]. The major question is then the existence of any (or just finite) local realization G of H, see [4] for the nonexistence and [2], [3] for the existence of algorithms.…”

“…Sedláček [18] introduced an N 2 -local property. The edge-induced subgraph on the set of all edges of a graph G that are adjacent to a given vertex x is denoted by N 2 (x, G).…”

Wheel-free planar graphs

“…Theorem 3.1 (Sedlaček [11]) If H is a graph of maximum degree at most three, then L(H) is planar if and only if H is planar.…”

On graphs with nonisomorphic $2$-neighbourhoods

“…He studied the class e ig l of connected graphs G with the fol lowing property: If x and y are two vertices of G, then N^x, G) and Ni(y, G) are not isomorphic. He proved Theorem 1.1 [7]. For every positive integer m ^ 6 there exists a graph G on m vertices belonging to ^Pj.…”

“…We have studied the generalization of the Trahtenbrot-Zykov problem to the jneighbourhoods, for j ^ i, [2]. Another direction of research was proposed by J. Sedte&k [7] in 1979. He studied the class e ig l of connected graphs G with the fol lowing property: If x and y are two vertices of G, then N^x, G) and Ni(y, G) are not isomorphic.…”