Decomposing a simple polygon into pseudo-triangles and convex polygons

In this paper we consider the problem of decomposing a simple polygon into subpolygons that exclusively use vertices of the given polygon. We allow two types of subpolygons: pseudo-triangles and convex polygons. We call the resulting decomposition PT-convex. We are interested in minimum decompositions, i.e., in decomposing the input polygon into the least number of subpolygons. Allowing subpolygons of one of two types has the potential to reduce the complexity of the resulting decomposition considerably.The problem of decomposing a simple polygon into the least number of convex polygons has been considered. We extend a dynamic-programming algorithm of Keil and Snoeyink for that problem to the case that both convex polygons and pseudo-triangles are allowed. Our algorithm determines such a decomposition in O(n³) time and space, where n is the number of the vertices of the polygon.     

A linear time algorithm to remove winding of a simple polygon

In this paper, we present a linear time algorithm to remove winding of a simple polygon P with respect to a given point q inside P. The algorithm removes winding by locating a subset of Jordan sequence that is in the proper order and uses only one stack.     

On the clique number of the square of a line graph and its relation to maximum degree of the line graph

In 1985, Erdős and Nešetřil conjectured that the square of the line graph of a graph , that is, , can be colored with colors. This conjecture implies the weaker conjecture that the clique number of such a graph, that is, , is at most . In 2015, Śleszyńska-Nowak proved that . In this paper, we prove that . This theorem follows from our stronger result that where .     

Solving the maximum clique problem using a tabu search approach

We describe two variants of a tabu search heuristic, a deterministic one and a probabilistic one, for the maximum clique problem. This heuristic may be viewed as a natural alternative implementation of tabu search for this problem when compared to existing ones. We also present a new random graph generator, the -generator, which produces graphs with larger clique sizes than comparable ones obtained by classical random graph generating techniques. Computational results on a large set of test problems randomly generated with this new generator are reported and compared with those of other approximate methods.The authors are grateful to the Quebec Government (Fonds F.C.A.R.) and to the Canadian Natural Sciences and Engineering Research Council (grant 0GP0038816) for financial support.     

Some remarks on the geodetic number of a graph

A set of vertices D of a graph G is geodetic if every vertex of G lies on a shortest path between two not necessarily distinct vertices in D. The geodetic number of G is the minimum cardinality of a geodetic set of G.We prove that it is NP-complete to decide for a given chordal or chordal bipartite graph G and a given integer k whether G has a geodetic set of cardinality at most k. Furthermore, we prove an upper bound on the geodetic number of graphs without short cycles and study the geodetic number of cographs, split graphs, and unit interval graphs.     

Cyclic hamiltonian cycle systems of the complete graph minus a 1-factor

In this paper, we prove that cyclic hamiltonian cycle systems of the complete graph minus a 1-factor, K_n-I, exist if and only if and n≠2p^α with p an odd prime and α?1.     

The maximum genus, matchings and the cycle space of a graph

In this paper we determine the maximum genus of a graph by using the matching number of the intersection graph of a basis of its cycle space. Our result is a common generalization of a theorem of Glukhov [5] and a theorem of Nebeský [15].     

Maximizing the expected number of components in an online search of a graph

The following optimal stopping problem is considered. The vertices of a graph G are revealed one by one, in a random order, to a selector. He aims to stop this process at a time t that maximizes the expected number of connected components in the graph ${\stackrel{?}{G}}_t$, induced by the currently revealed vertices. The selector knows G in advance, but different versions of the game are considered depending on the information that he gets about ${\stackrel{?}{G}}_t$. We show that when G has N vertices and maximum degree of order $o(\sqrt{N})$, then the number of components of ${\stackrel{?}{G}}_t$ is concentrated around its mean, which implies that playing the optimal strategy the selector does not benefit much by receiving more information about ${\stackrel{?}{G}}_t$. Results of similar nature were previously obtained by M. Lasoń for the case where G is a k-tree (for constant k). We also consider the particular cases where G is a square, triangular or hexagonal lattice, showing that an optimal selector gains cN components and we compute c with an error less than 0.005 in each case.     

On the number of noncritical vertices in strongly connected digraphs

By definition, a vertex w of a strongly connected (or, simply, strong) digraph D is noncritical if the subgraph D — w is also strongly connected. We prove that if the minimal out (or in) degree k of D is at least 2, then there are at least k noncritical vertices in D. In contrast to the case of undirected graphs, this bound cannot be sharpened, for a given k, even for digraphs of large order. Moreover, we show that if the valency of any vertex of a strong digraph of order n is at least 3/4n, then it contains at least two noncritical vertices. The proof makes use of the results of the theory of maximal proper strong subgraphs established by Mader and developed by the present author. We also construct a counterpart of this theory for biconnected (undirected) graphs.     

Bounding the crossing number of a graph in terms of the crossing number of a minor with small maximum degree

We show that if G has a minor M with maximum degree at most 4, then the crossing number of G in a surface Σ is at least one fourth the crossing number of M in Σ. We use this result to show that every graph embedded on the torus with representativity r ≥ 6 has Klein bottle crossing number at least ⌊2r/3⌋²/64. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 168–173, 2001     

The maximum multiplicity of the largest k-th eigenvalue in a matrix whose graph is acyclic or unicyclic

Given a graph $G$ we are interested in studying the symmetric matrices associated to $G$ with a fixed number of negative eigenvalues. For this class of matrices we focus on the maximum possible nullity. For trees this parameter has already been studied and plenty of applications are known. In this work we derive a formula for the maximum nullity and completely describe its behavior as a function of the number of negative eigenvalues. In addition, we also carefully describe the matrices associated with trees that attain this maximum nullity. The analysis is then extended to the more general class of unicyclic graphs. Further our work is applied to re-describing all possible partial inertias associated with trees, and is employed to study an instance of the inverse eigenvalue problem for certain trees.