Reliability analysis of exchanged hypercubes based on the path connectivity

Let G be a connected simple graph with vertex set V(G) and edge set E(G). For any subset eta of V(G) with |eta| >= 2, let pi(G)(eta) denote the maximum number t of paths P-1, P-2, . . . , P-t in G such that Picontains eta, V(Pi) boolean AND V(Pj) = eta and E(P-i) boolean AND E(P-j) = empty set for any distinct i, j is an element of {1, 2, . . . , t}. For an integer k with 2 <= k <= |V(G)|, the k-path connectivity pi(k)(G) of G, which can more accurately assess the reliability of networks, is defined as min{pi G(eta)|eta subset of V(G) and |eta| = k}. Since deciding whether pi(G)(eta) >= l for a general graph is NP-complete in [Graphs Combin. 37(2021)2521-2533], there are few results about k-path connectivity even for k = 3. In this paper, we obtain the exact value of the 3-path connectivity of the exchanged hypercube EH(s, t) and show that pi(3)(EH(s, t)) = left perpendicular 3xmin{s,t}+2/4 right perpendicular which improves the known result about the 3-tree connectivity [Appl. Math. Comput. 347(2019)342-353]. As a corollary, the 3-path connectivity of the n-dimensional dual cube D-n is obtained directly. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.