On Γ-regular graphs

The subclass of directed series-parallel graphs plays an important role in computer science. To determine whether a graph is series-parallel is a well studied problem in algorithmic graph theory. Fast sequential and parallel algorithms for this problem have been developed in a sequence of papers. For series-parallel graphs methods are also known to solve the reachability and the decomposition problem time efficiently. However, no dedicated results have been obtained for the space complexity of these problems - the topic of this paper. For this special class of graphs, we develop deterministic algorithms for the recognition, reachability, decomposition and the path counting problem that use only logarithmic space. Since for arbitrary directed graphs reachability and path counting are believed not to be solvable in log-space the main contribution of this work are novel deterministic path finding routines that work correctly in series-parallel graphs, and a characterisation of series-parallel graphs by forbidden subgraphs that can be tested space-efficiently. The space bounds are best possible, i.e. the decision problems is shown to be ζ-complete with respect to AC 0 -reductions, and they have also implications for the parallel time complexity of series-parallel graphs. Finally, we sketch how these results can be generalised to extension of the series-parallel graph family: to graphs with multiple sources or multiple sinks and to the class of minimal vertex series-parallel graphs.

Space efficient algorithms for series-parallel graphs

Structure fault tolerance of k-ary n-cube networks

Packing internally disjoint Steiner trees to compute the κ3-connectivity in augmented cubes

We consider the problem of determining the possible orders for k-regular, k-connected and bipancyclic subgraphs of the hypercube Qn. For k = 2 and k = 3, the solution to the problem is known. In this paper, we solve the problem for k = 4 by proving that Qn has a 4-regular, 4-connected and bipancyclic subgraph on l vertices if and only if l = 16 or l is an even integer such that 24 ≤ l ≤ 2n. Further, by improving a result of Ramras, we prove that a k-regular subgraph of Qn is either isomorphic to Qk or has at least 2k + 2k−1 vertices. We also improve a result of Mane and Waphare regarding the existence of a k-regular, k-connected and bipancyclic subgraph of Qn. Some applications of our results are given.

On 4-regular 4-connected bipancyclic subgraphs of hypercubes

In this paper, we consider the problem of decomposing the edge set of the hypercube Qn into two spanning, regular, connected, bipancyclic subgraphs. We prove that if n = n1 + n2 with n1 ≥ 2 and n2 ≥ 2, then the edge set of Qn can be decomposed into two spanning, bipancyclic subgraphs H1 and H2 such that Hi is ni-regular and ni-connected for i = 1, 2.

Decomposition of hypercubes into regular connected bipancyclic subgraphs

Abstract An m-factorization of a graph is a decomposition of its edge set into edge-disjoint m-regular spanning subgraphs (or factors). In this paper, we prove the existence of an isomorphic m-factorization of the Cartesian product of graphs each of which is decomposable into Hamiltonian even cycles. Moreover, each factor in the m-factorization is m-connected, and bipancyclic for m ≥ 4 and nearly bipancyclic for m = 3.

/pdf/connected-bipancyclic-isomorphic-m-factorizations-of-the-3neygoieiy.pdf

Connected bipancyclic isomorphic m-factorizations of the Cartesian product of graphs.

On n-connected splitting matroids

It is known that the n-dimensional hypercube Qn, for n = n1 + n2 with ni ≥ 2, can be decomposed into two spanning bipancyclic subgraphs G1 and G2 such that Gi is ni-regular and ni-connected for i = 1, 2. In this paper, we prove that if n = n1 + n2 + ⋯ + nk with ni ≥ 2 and at most one ni odd, then Qn can be decomposed into k spanning subgraphs G1, G2,…,Gk such that Gi is ni-regular and ni-connected for i = 1, 2,…,k.

Y. M. Borse

Papers

Decomposition of hypercubes into regular connected bipancyclic subgraphs

On 4-regular 4-connected bipancyclic subgraphs of hypercubes

Connected bipancyclic isomorphic m-factorizations of the Cartesian product of graphs.

On n-connected splitting matroids

Decomposing hypercubes into regular connected subgraphs