Sufficient conditions for λ′-optimality of graphs with small conditional diameter

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

“…Furthermore, a λ (2) -connected graph is said to be λ (2) -optimal if λ (2) = ξ. Recent results on this property are obtained in [2,5,12,13,18,21,23]. Notice that if λ (2) ≤ δ, then λ (2) = λ.…”

“…Therefore, by means of this parameter we can say that a graph G is super-λ if and only if λ (2) > δ. Thus, we can measure the super edge-connectivity of the graph as the value of the restricted edge-connectivity λ (2) .…”

“…And Figure 2 shows a 3-regular graph G with λ(G) = κ(G) = 2, and its 3-arc graph X(G) which has λ(X(G)) = 4 and λ (2) (X(G)) = 6 = ξ(X(G)), i.e., this graph is λ (2) -optimal. In what follows we give a lower bound on the restricted edge-connectivity λ (2) (X(G)) where G is a graph having connectivity κ(G) ≥ 2.…”

“…Then X(G) has restricted edge-connectivity λ (2) Figure 3 shows that λ(G) ≥ 2 is not enough to guarantee that λ (2) (X(G)) ≥ 2(δ − 1) 2 − 2. In this example G is a 4-regular graph with λ = 2 and κ = 1, but λ (2) (X(G)) = 12 < 16.…”

“…The restricted edgeconnectivity λ (2) = λ (2) (G) is the minimum cardinality over all restricted edge-cuts W , i.e., those such that there are no isolated vertices in G − W . A restricted edge-cut W is called a λ (2) -cut if |W | = λ (2) . Obviously for any λ (2) -cut W , the graph G − W consists of exactly two components On the connectivity and restricted edge-connectivity of 3-arc graphs C. Balbuena et al…”