On the edge-to-vertex detour number of a graph (Q2923328)

For a graph \(G=(V,E)\), a set \(S\subseteq E\) is called an edge-to-vertex detour set if every vertex of \(G\) is incident with an edge of \(S\) or lies on a detour joining a pair of edges of \(S\). The edge-to-vertex detour number \(dn_2(G)\) of \(G\) is the minimum order of its edge-to-vertex detour sets. In this paper, the graphs \(G\) of size \(q\) for which \(dn_2(G)=q\) or \(dn_2(G)=q-1\) or \(dn_2(G)=q-2\) are characterized. The following theorems are also proved. NEWLINENEWLINENEWLINENEWLINE Let \(G\) be a connected graph of size \(q\geq 2\). Then \(dn_2(G)=q\) if and only if \(G\) is a star \(K_{1,q}\). NEWLINENEWLINENEWLINE Let \(G\) be a connected graph of size \(q\geq 3\). Then \(dn_2(G)=q-1\) if and only if \(G\) is \(K_3\) or a double star. NEWLINENEWLINENEWLINE Let \(G\) be a connected graph of size \(q\geq 4\). Then \(dn_2(G)=q-2\) if and only if \(G\) is \(C_4\) or \(K_{1,q-1}+e\) or a caterpillar of detour diameter 4.