We present eecient polynomial time algorithms that place bn=2c vertex guards which cover the surface of an n-vertex polyhedral terrain, and similarly, bn=3c edge guards which cover the surface of an n-vertex polyhedral terrain. The time complexity of both algorithms, dominated by the cost of nding a maximum matching in a graph, is O(n 3=2).

An arborescence graph is a directed in which, for vertex u called the root, and any other v, there exactly one path from to v. The pathos of an Ar defined as collection minimum number arc disjoint open paths whose union Ar. In [6], Ar, total digraph Q = DP T(Ar)has set V (Q) (Ar) ∪ A(Ar) P(Ar), where set, P(Ar) A(Q) consists following arcs: ab such that a, b ∈ head coincides with tail b; uv u, ...

OBJECTIVE To establish reference range for uterine artery (UtA) Doppler pulsatility index (PI) using transvaginal ultrasound at 20-24w6d of gestation in a Brazilian population. MATERIAL AND METHODS A retrospective cross-sectional study in 847 low-risk pregnant women undergoing routine second trimester ultrasound examination was conducted from February 2012 through March 2015. The mean UtA PI ...

We define a higher spin alternating sign matrix to be an integer-entry square matrix in which, for a nonnegative integer r, all complete row and column sums are r, and all partial row and column sums extending from each end of the row or column are nonnegative. Such matrices correspond to configurations of spin r/2 statistical mechanical vertex models with domain-wall boundary conditions. The c...

Vertex Cover is one of the most well studied problems in the realm of parameterized algorithms and admits a kernel with O(`2) edges and 2` vertices. Here, ` denotes the size of a vertex cover we are seeking for. A natural question is whether Vertex Cover admits a polynomial kernel (or a parameterized algorithm) with respect to a parameter k, that is, provably smaller than the size of the vertex...

Given an n-vertex graph G and a function f : V (G)→ {0, . . . , n−1}, an f -factor is a subgraph H of G such that degH(v) = f(v) for every vertex v ∈ V (G); we say that H is a connected f -factor if, in addition, the subgraph H is connected. A classical result of Tutte (1954) is the polynomial time algorithm to check whether a given graph has a specified f -factor. However, checking for the pre...

We prove that if an edge of a cycle on n vertices is extended by adding k vertices, then the the chromatic polynomial of such generalized cycle is: P (Hk, λ) = (λ− 1) k ∑ i=0 λ + (−1)n(λ− 1).

It is shown that the compressed word problem of a graph product of finitely generated groups is polynomial time Turing-reducible to the compressed word problems of the vertex groups. A direct corollary of this result is that the word problem for the automorphism group of a right-angled Artin group or a right-angled Coxeter group can be solved in polynomial time. Moreover, it is shown that a res...

A generalized vertex join of a graph is obtained by joining an arbitrary multiset of its vertices to a new vertex. We present a low-order polynomial time algorithm for finding the chromatic polynomials of generalized vertex joins of trees, and by duality we find the flow polynomials of arbitrary outerplanar graphs. We also present closed formulas for the chromatic and flow polynomials of vertex...

A graph is d-degenerate if its every subgraph contains a vertex of degree at most d. For instance, planar graphs are 5-degenerate. Inspired by recent work by Philip, Raman and Sikdar, who have shown the existence of a polynomial kernel for DOMINATING SET in d-degenerate graphs, we investigate kernelization hardness of problems that include connectivity requirement in this class of graphs. Our m...