\(N_ 2\)-locally disconnected graphs (Q1309470)

A new type of neighbourhood of a vertex in a graph is introduced. An edge \(yz\) of an undirected graph \(G\) is called adjacent to a vertex \(x\), if \(y \neq x \neq z\) and at least one of the vertices \(y\), \(z\) is adjacent to \(x\). The neighbourhood of the second type of \(x\) is the edge-induced subgraph of \(G\) on the edges that are adjacent to \(x\); it is denoted by \(N_ 2(x,G)\). If \(N_ 2(x,G)\) is disconnected for each vertex \(x\) of \(G\), then \(G\) is called \(N_ 2\)-locally disconnected. For \(n\geq 5\), by \(t(n)\) (or \(t_ p(n))\) the maximum number of edges of an \(N_ 2\)- locally disconnected (or planar \(N_ 2\)-locally disconnected, respectively) graph with \(n\) vertices is denoted. Exact values of \(t_ p(n)\) are found for all \(n \geq 5\). For \(t(n)\) both the lower bound and the upper bound are determined.