The Complexity of Reliability Computations in Planar and Acyclic Graphs

We show that the problem of computing source-sink reliability is NP-hard, in fact # P-complete, even for undirected and acyclic directed source-sink planar graphs having vertex degree at most three. Thus the source-sink reliability problem is unlikely to have an efficient algorithm, even when the graph can be laid out on a rectilinear grid

Parity balance of the ii-th dimension edges in Hamiltonian cycles of the hypercube

Let $n\geq 2$ be an integer, and let $i\in\{0,...,n-1\}$. An $i$-th dimension edge in the $n$-dimensional hypercube $Q_n$ is an edge ${v_1}{v_2}$ such that $v_1,v_2$ differ just at their $i$-th entries. The parity of an $i$-th dimension edge \edg{v_1}{v_2} is the number of 1's modulus 2 of any of its vertex ignoring the $i$-th entry. We prove that the number of $i$-th dimension edges appearing in a given Hamiltonian cycle of $Q_n$ with parity zero coincides with the number of edges with parity one. As an application of this result it is introduced and explored the conjecture of the inscribed squares in Hamiltonian cycles of the hypercube: Any Hamiltonian cycle in $Q_n$ contains two opposite edges in a 4-cycle. We prove this conjecture for $n \le 7$, and for any Hamiltonian cycle containing more than $2^{n-2}$ edges in the same dimension. This bound is finally improved considering the equi-independence number of $Q_{n-1}$, which is a concept introduced in this paper for bipartite graphs

Parity results on connected ƒ-factors

AbstractLet G be a connected graph with vertex set V and let d(ν) denote the degree of a vertex νϵV. For ƒ a mapping from V to the positive integers, an ƒ-factor is a spanning subgraph having degree ƒ(ν) at vertex ν. In this paper we extend the parity results of Thomason [2] on Hamiltonian circuits to connected ƒ-factors. (A Hamiltonian circuit is a connected 2-factor.) We show that if ƒ(ν) and d(ν) have opposite parity for all νϵV then for any given subgraph C there is an even number of connected ƒ-factors having C as a cotree.Let ƒ1 and ƒ2 be any mappings from V to the positive integers that partition d, i.e., d(ν) = ƒ1(ν) +ƒ2(ν) for all νϵV. Let C1 and C2 be any pair of edge disjoint subgraphs. We also show in this paper that the number of decompositions of G into a connected ƒ1-factor having C1 as a cotree and a connected ƒ2-factor having C2 as a cotree is even