Discrete Applied Mathematics

In Hartnell's firefighter game a player tries to contain a fire breaking out at some vertex of a graph and spreading in rounds from burned vertices to their neighbors, by defending one vertex in each round, which will remain protected from the fire ...

In this work we present an approach for designing the lines of a rapid transit network. Given the stations to be constructed and the links between them, a set of lines is generated by utilizing a greedy heuristic procedure that, taking into account the ...

For positive integers k and d such that 4 ź k < d and k ź 5 , we determine the maximum number of rainbow colored copies of C 4 in a k -edge-coloring of the d -dimensional hypercube Q d . Interestingly, the k -edge-colorings of Q d yielding the maximum ...

A cycle transversal or feedback vertex set of a graph G is a subset T ź V ( G ) such that T ź V ( C ) ź 0ź for every cycle C of G . A clique cycle transversal, or cct for short, is a cycle transversal which is a clique. Recognizing graphs which admit a ...

Satisfying spin-assignments of triangulations of a surface are states of minimum energy of the antiferromagnetic Ising model on triangulations which correspond (via geometric duality) to perfect matchings in cubic bridgeless graphs. In this work we show ...

We prove that a stack of n pancakes is rearranged in all n ! ways by repeatedly applying the following rule: Flip the maximum number of pancakes that gives a new stack. This complements the previously known pancake flipping Gray code (Zaks, 1984) which ...

A repetition-free Longest Common Subsequence (LCS) of two sequences x and y is an LCS of x andź y where each symbol may appear at most once. Let R denote the length of a repetition-free LCS of two sequences of n symbols each one chosen randomly, ...

We consider the problem of finding the point in the transportation polytope which is closest to the origin. Recursive formulas to solve it are provided, explaining how they arise from geometric considerations, via projections, and we derive solution ...

Graph searching games involve a team of searchers that aims at capturing a fugitive in a graph. These games have been widely studied for their relationships with the tree- and the path-decomposition of graphs. In order to define decompositions for ...

Perfect matchings and maximum weight matchings are two fundamental combinatorial structures. We consider the ratio between the maximum weight of a perfect matching and the maximum weight of a general matching. Motivated by the computer graphics ...

We prove a conjecture of Aanjaneya, Bishnu, and Pal that the minimum number of diffuse reflections sufficient to illuminate the interior of any simple polygon with n walls from any interior point light source is ź n / 2 ź - 1 . Light reflecting ...

Given a multigraphź G and a functionź F that assigns a forbidden ordered pair of colors to each edge e , we say a coloringź C of the vertices is conforming to ź F if for all e = ( u , v ) , ( C ( u ) , C ( v ) ) ź F ( e ) . Conforming colorings ...

An effective way to analyze and apprehend the structural properties of networks is to find their most critical nodes. This makes them easier to control, whether the purpose is to keep or to delete them. Given a graph, Critical Node Detection problem (...

The natural extension of the concept of perfection in graphs to hypergraphs is to define a uniform m -hypergraph, H , as perfect, if it satisfies that for every subhypergraph H ' , ź ( H ' ) = ź ω ( H ' ) m - 1 ź , where ź ( H ' ) and ω ( H ' ) are the ...

In this paper, we study the (geodesic) hull number of graphs. For any two vertices u , v ź V of a connected undirected graph G = ( V , E ) , the closed interval I u , v of u and v is the set of vertices that belong to some shortest ( u , v ) -path. For ...

In this paper we study lift-and-project polyhedral operators defined by Lovász and Schrijver and Balas, Ceria and Cornuéjols on the clique relaxation of the stable set polytope of webs. We compute the disjunctive rank of all webs and consequently of ...

In this paper we continue the study of the edge intersection graphs of one (or zero) bend paths on a rectangular grid. That is, the edge intersection graphs where each vertex is represented by one of the following shapes: źź , ź , źß , ź , and we ...

By considering graphs as topological spaces we introduce, at the level of homology, the notion of a null coloring, which provides new information on the task of clarifying the structure of cycles in a graph. We prove that for any graph G a maximal null ...

Let r ( G , H ) be the smallest integer N such that for any 2 -coloring (say, red and blue) of the edges of K n , n ź N , there is either a red copy of G or a blue copy of H . Let K n - K 1 , s be the complete graph on n vertices from which the edges of ...

For positive integers c , s ź 1 , let M 3 ( c , s ) be the least integer such that any set of at least M 3 ( c , s ) points in the plane, no three on a line and colored with c colors, contains a monochromatic triangle with at most s interior points. The ...

We study the set covering polyhedron related to circulant matrices. In particular, our goal is to characterize the first Chvátal closure of the usual fractional relaxation. We present a family of valid inequalities that generalizes the family of minor ...

Given a graph G = ( V , E ) , a family of nonempty vertex-subsets S ź 2 V , and a weight w : S ź R + , the maximum stable set problem with weights on vertex-subsets consists in finding a stable set I of G maximizing the sum of the weights of the sets in ...

A kernel N of a digraph D is an independent set of vertices of D such that for every w ź V ( D ) ź N there exists an arc from w to N . The digraph D is said to be a kernel-perfect digraph when every induced subdigraph of D has a kernel. Minimal non ...

In this paper we consider the Robust Connected Facility Location (ConFL) problem within the robust discrete optimization framework introduced by Bertsimas and Sim (2003). We propose an Approximate Robust Optimization (ARO) method that uses a heuristic ...

Basic chordal graphs arose when comparing clique trees of chordal graphs and compatible trees of dually chordal graphs. They were defined as those chordal graphs whose clique trees are exactly the compatible trees of its clique graph.In this work, we ...

The Radon number of a graph is the minimum integer r such that all sets of at least r of its vertices can be partitioned into two subsets whose convex hulls intersect. Determining the Radon number of general graphs in the geodetic convexity is NP-hard. ...

One wishes to remove k - 1 edges of a vertex-weighted tree T such that the weights of the k induced connected components are approximately the same. How well can one do it? In this paper, we investigate such k -separators for quasi-binary trees. We show ...

The middle levels problem consists in determining a hamiltonian cycle in the bipartite Kneser graph B ( 2 k + 1 , k ) , also known as the middle levels graph and denoted by B k . Previously, it was proved that a particular hamiltonian path in a reduced ...