Sparse Symmetric Positive Definite (SPD) Linear System Solvers (LSS) are routinely required in various Power System Analysis problems. In sparse LSS ordering of nodes is desired to reduce fill ins, which in turn reduce computations. Minimum Degree Algorithm (MDA) has established itself as a very competitive ordering algorithm, for reducing fill ins to quite often near minimum level. With the ac...

It is well known that the set of possible degree sequences for a simple graph on n vertices is the intersection of a lattice and a convex polytope. We show that the set of possible degree sequences for a simple k-uniform hypergraph on n vertices is not the intersection of a lattice and a convex polytope for k > 3 and n > k + 13. We also show an analogous nonconvexity result for the set of degre...

For a given m, 0 < m ≤ 2, let Dm(n) denote the set of all hypergraphic sequences for hypergraphs with n vertices and m hyperedges. A hypergraphic sequence in Dm(n) is upper hypergraphic if all its components are at least m/2. Let ?̂?m(n) denote the set of all upper hypergraphic sequences. A structural characterization of the lowest and highest rank maximal elements of ?̂?m(n) was provided in an e...

We study approximation hardness of the Minimum Dominating Set problem and its variants in undirected and directed graphs. Using a similar result obtained by Trevisan for Minimum Set Cover we prove the first explicit approximation lower bounds for various kinds of domination problems (connected, total, independent) in bounded degree graphs. Asymptotically, for degree bound approaching infinity, ...

let $g$ be a connected graph with vertex set $v(g)$‎. ‎the‎ ‎degree resistance distance of $g$ is defined as $d_r(g) = sum_{{u‎,‎v} subseteq v(g)} [d(u)+d(v)] r(u,v)$‎, ‎where $d(u)$ is the degree‎ ‎of vertex $u$‎, ‎and $r(u,v)$ denotes the resistance distance between‎ ‎$u$ and $v$‎. ‎in this paper‎, ‎we characterize $n$-vertex unicyclic‎ ‎graphs having minimum and second minimum degree resista...

We present a simple algorithm for the maximum independent set problem in an n-vertex graph with degree bounded by 4, which runs in O∗(1.1455n) time and improves all previous algorithms for this problem. In this paper, we use the “Measure and Conquer method” to analyze the running time bound, and use some good reduction and branching rules with a new idea on setting weights to obtain the improve...

Halldórsson (1995) claimed a (k + 1)/3 + 2 performance ratio for approximating the degree-bounded k-set packing problem. However, its detailed proof was omitted therein. In this report, we attempt to present in detail the proof toward this performance ratio which was briefly given in Halldórsson and Radhakrishnan (1994), and further show that the underlying local improvement search algorithm ru...

We consider the problem of nding an independent dominating set of minimum cardinality in bounded degree and regular graphs. We rst give an approximation algorithm for at most cubic graphs, that achieves ratio 2, based on greedy and local search techniques. We then propose an heuristic based on an iterative application of the greedy technique on graphs of lower and lower degree. When the graph i...

Rough set theory is a relatively new mathematical tool for use in computer applications in circumstances which are characterized by vagueness and uncertainty. In this paper, applications of inclusion degree in rough set theory are discussed, the relationships among inclusion degree, measures on rough set theory and some generalized rough set models are established. These results will be very he...