Discrete Applied Mathematics

An L(p,q)-labeling of a graph G is an assignment f from vertices of G to the set of non-negative integers {0,1,...,@l} such that |f(u)-f(v)|>=p if u and v are adjacent, and |f(u)-f(v)|>=q if u and v are at distance 2 apart. The minimum value of @l for ...

In this paper, we obtain several tight bounds of the defensive k-alliance number in the complement graph from other parameters of the graph. In particular, we investigate the relationship between the alliance numbers of the complement graph and the ...

Several important combinatorial arrays, after inserting some minus signs, turn out to be involutions when considered as lower triangular matrices. Among these are the Pascal, RNA, and directed animal matrices. These examples and many others are in the ...

We consider a combinatorial problem motivated by a special simplified timetabling problem for subway networks. Mathematically the problem is to find (pairwise) disjoint congruence classes modulo certain given integers; each such class corresponds to the ...

An undirected graph is a treelike comparability graph if it admits a transitive orientation such that its transitive reduction is a tree. We show that treelike comparability graphs are distance hereditary. Utilizing this property, we give a linear time ...

The main goal of coding theory is to devise efficient systems to exploit the full capacity of a communication channel, thus achieving an arbitrarily small error probability. Low Density Parity Check (LDPC) codes are a family of block codes-characterised ...

For two vertices u and v in a strong oriented graph D, the strong distance sd(u,v) between u and v is the minimum size (the number of arcs) of a strong sub-digraph of D containing u and v. For a vertex v of D, the strong eccentricity se(v) is the strong ...

We present a new representation of a chordal graph called the clique-separator graph, whose nodes are the maximal cliques and minimal vertex separators of the graph. We present structural properties of the clique-separator graph and additional ...

Given a graph G and two positive integers p,q with p>q an L(p,q)-labeling of G is a function f from the vertex set V(G) to the set of all nonnegative integers such that |f(x)-f(y)|>=p if d"G(x,y)=1 and |f(x)-f(y)|>=q if d"G(x,y)=2. A k-L(p,q)-labeling ...

A (k,g)-cage is a k-regular graph of girth g and with the least possible number of vertices. In this paper, we investigate the problem of how many connected components there will be after removing a cutset of up to k vertices from a (k,g)-cage.

The Randic index of a graph G is defined as R(G)=@?"u"~"v(d(u)d(v))^-^1^2, where d(u) is the degree of vertex u and the summation goes over all pairs of adjacent vertices u, v. A conjecture on R(G) for connected graph G is as follows: R(G)>=r(G)-1, ...

In this paper we analyze several approaches to the Maximum Independent Set (MIS) problem in hypergraphs with degree bounded by a parameter @D. Since independent sets in hypergraphs can be strong and weak, we denote by MIS (MSIS) the problem of finding a ...

A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes. El-Sahili and Kouider have conjectured that every d-regular graph with girth at least 5 has a b-coloring ...

The tenacity of a graph G, T(G), is defined by T(G)=min{|S|+@t(G-S)@w(G-S)}, where the minimum is taken over all vertex cutsets S of V(G), @w(G-S) be the number of components of G-S and @t(G-S) be the number of vertices in the largest component of the ...

An (abstract) convex geometry is a combinatorial abstraction of convexity which is a Moore family with the closure operator satisfying the anti-exchange property. A number of results of matroids on the NBC-complexes (or broken circuit complexes) happen ...

The Economic Order Quantity (EOQ) problem is a fundamental problem in supply and inventory management. In its classical setting, solutions are not affected by the warehouse capacity. We study a type of EOQ problem where the (maximum) warehouse capacity ...

In this paper we consider the problem of scheduling jobs with release dates on parallel unbounded batch processing machines to minimize the maximum lateness. We show that the case where the jobs have deadlines is strongly NP-hard. We develop a ...

In this paper we introduce a new class of clustering problems. These are similar to certain classical problems but involve a novel combination of @?"p-statistics and @?"q norms. We discuss a real world application in which the case p=2 and q=1 arises in ...

Given a graph with edge costs, the power of a node is the maximum cost of an edge leaving it, and the power of a graph is the sum of the powers of its nodes. Motivated by applications in wireless networks, we consider several fundamental undirected ...

The left-regular multiplication is explicitly embedded in the notion of perfect nonlinearity. But there exist many other group actions. By replacing translations by another group action the new concept of group action-based perfect nonlinearity has been ...

Efficient data gathering is an important challenge in sensor networks. In this paper we address the problem of gathering sensed data to the sink of a sensor network minimizing the time to complete the process. We present optimal time data gathering ...

The fractional weak discrepancywd"F(P) of a poset P=(V,@__ __) was introduced in [A. Shuchat, R. Shull, A. Trenk, The fractional weak discrepancy of a partially ordered set, Discrete Applied Mathematics 155 (2007) 2227-2235] as the minimum nonnegative k ...

It has been known that every planar 4-graph has a 2-bend 2-D orthogonal drawing, with the only exception being the octahedron, every planar 3-graph has a 1-bend 2-D orthogonal drawing with the only exception being K"4, and every outerplanar 3-graph with ...

Let j>=k>=0 be integers. An @?-L(j,k)-labelling of a graph G=(V,E) is a mapping @f:V->{0,1,2,...,@?} such that |@f(u)-@f(v)|>=j if u,v are adjacent and |@f(u)-@f(v)|>=k if they are distance two apart. Let @l"j","k(G) be the smallest integer @? such that ...

Let G be a finite and simple graph with vertex set V(G), and let f:V(G)->{-1,1} be a two-valued function. If @?"x"@?"N"["v"]f(x)>=1 for each v@?V(G), where N[v] is the closed neighborhood of v, then f is a signed dominating function on G. A set {f"1,f"2,...

In traditional edge searching one tries to clean all of the edges in a graph employing the least number of searchers. It is assumed that each edge of the graph initially has a weight equal to one. In this paper we modify the problem and introduce the ...

A defensive alliance in a graph G=(V,E) is a set of vertices S@?V satisfying the condition that, for each v@?S, at least one half of its closed neighbors are in S. A defensive alliance S is called a critical defensive alliance if any vertex is removed ...

Assume we have a set of k colors and we assign an arbitrary subset of these colors to each vertex of a graph G. If we require that each vertex to which an empty set is assigned has in its neighborhood all k colors, then this assignment is called the k-...