On the connectivity and restricted edge-connectivity of 3-arc graphs

• Authors: C. Balbuena;P. García-Vázquez;L. P. Montejano
• Venue: Discrete Applied Mathematics
• Year: 2014

Let G@? denote the symmetric digraph of a graph G. A 3-arc is a 4-tuple (y,a,b,x) of vertices such that both (y,a,b) and (a,b,x) are paths of length 2 in G. The 3-arc graphX(G) of a given graph G is defined to have vertices the arcs of G@?, and they are denoted as (uv). Two vertices (ay),(bx) are adjacent in X(G) if and only if (y,a,b,x) is a 3-arc of G. The purpose of this work is to study the edge-connectivity and restricted edge-connectivity of 3-arc graphs. We prove that the 3-arc graph X(G) of every connected graph G of minimum degree @d(G)=3 has @l(X(G))=(@d(G)-1)^2. Furthermore, if G is a 2-connected graph, then X(G) has restricted edge-connectivity @l"("2")(X(G))=2(@d(G)-1)^2-2. We also provide examples showing that all these bounds are sharp. Concerning the vertex-connectivity, we prove that @k(X(G))=min{@k(G)(@d(G)-1),(@d(G)-1)^2}. This result improves a previous one by [M. Knor, S. Zhou, Diameter and connectivity of 3-arc graphs, Discrete Math. 310 (2010) 37-42]. Finally, we obtain that X(G) is superconnected if G is maximally connected.