Special Digraph Research Articles

Weak-odd chromatic index of special digraph classes

Given a digraph D = ( V ( D ) , A ( D ) ) , let ∂ D + ( v ) = { v w | w ∈ N D + ( v ) } and ∂ D − ( v ) = { u v | u ∈ N D − ( v ) } be semi-cuts of v . A mapping φ : A ( D ) → [ k ] is called a weak-odd k -edge coloring of D if it satisfies the condition: for each v ∈ V ( D ) , there is at least one color with an odd number of occurrences on each non-empty semi-cut of v . We call the minimum integer k the weak-odd chromatic index of D . When limit to 2 colors, let def ( D ) denote the defect of D , i.e the minimum number of vertices in D at which the above condition is not satisfied. In this paper, we give a descriptive characterization with respect to the weak-odd chromatic index and the defect of semicomplete digraphs and extended tournaments, which generalize results of tournaments to broader classes. In addition, we initiate the study of weak-odd edge covering on digraphs.

Complementary cycles of any length in regular bipartite tournaments

AbstractLet be a ‐regular bipartite tournament on vertices. We show that, for every with , has a cycle of length such that is Hamiltonian unless is isomorphic to the special digraph . This statement was conjectured by Zhang, Manoussakis and Song. In the same paper, the conjecture was proved for and more recently Bai, Li and He gave a proof for .

The majority coloring of the join and Cartesian product of some digraph

A majority coloring of a digraph is a vertex coloring such that for every vertex, the number of vertices with the same color in the out-neighborhood does not exceed half of its out-degree. Kreutzer, Oum, Seymour and van der Zyper proved that every digraph is majority 4-colorable and conjecture that every digraph has a majority 3-coloring. This paper mainly studies the majority coloring of the joint and Cartesian product of some special digraphs and proved the conjecture is true for the join graph and the Cartesian product. According to the influence of the number of vertices in digraph, we prove the majority coloring of the joint and Cartesian product of some digraph.

Duality respecting representations and compatible complexity measures for gammoids

We show that every gammoid has special digraph representations, such that a representation of the dual of the gammoid may be easily obtained by reversing all arcs. In an informal sense, the duality notion of a poset applied to the digraph of a special representation of a gammoid commutes with the operation of forming the dual of that gammoid. We use these special representations in order to define a complexity measure for gammoids, such that the classes of gammoids with bounded complexity are closed under duality, minors, and direct sums.

Solutions for subset sum problems with special digraph constraints

The subset sum problem is one of the simplest and most fundamental NP-hard problems in combinatorial optimization. We consider two extensions of this problem: The subset sum problem with digraph constraint (SSG) and subset sum problem with weak digraph constraint (SSGW). In both problems there is given a digraph with sizes assigned to the vertices. Within SSG we want to find a subset of vertices whose total size does not exceed a given capacity and which contains a vertex if at least one of its predecessors is part of the solution. Within SSGW we want to find a subset of vertices whose total size does not exceed a given capacity and which contains a vertex if all its predecessors are part of the solution. SSG and SSGW have been introduced recently by Gourvès et al. who studied their complexity for directed acyclic graphs and oriented trees. We show that both problems are NP-hard even on oriented co-graphs and minimal series-parallel digraphs. Further, we provide pseudo-polynomial solutions for SSG and SSGW with digraph constraints given by directed co-graphs and series-parallel digraphs.

The negative tetrahedron and the first infinite family of connected digraphs that are strongly determined by the Hermitian spectrum

Thus far, digraphs that are uniquely determined by their Hermitian spectra have proven elusive. Instead, researchers have turned to spectral determination of classes of switching equivalent digraphs, rather than individual digraphs. In the present paper, we consider the traditional notion: a digraph (or mixed graph) is said to be strongly determined by its Hermitian spectrum (abbreviated SHDS) if it is isomorphic to each digraph to which it is cospectral. Convincing numerical evidence to support the claim that this property is extremely rare is provided. Nonetheless, the first infinite family of connected digraphs that is SHDS is constructed. This family is obtained via the introduction of twin vertices into a structure that is named negative tetrahedron. This special digraph, that exhibits extreme spectral behavior, is contained in the surprisingly small collection of all digraphs with exactly one negative eigenvalue, which is determined as an intermediate result.

Magic and Modulo Magicness of Paley Digraph

A special digraph arises in round robin tournaments. More exactly, a tournament Tq with q players 1, 2, ... , q in which there are no draws. This gives rise to a digraph in which either (u, v) or (v, u) is an arc for each pair u, v. Graham and Spencer defined the tournament as, The nodes of digraph Dp are {0, 1, ... , p -1} and Dp contains the arc (u, v) if and only if u - v is a quadratic residue modulo p where p  3(mod 4) be a prime. This digraph is referred as the Paley tournament. Raymond Paley was a person raised Hadamard matrices by using this quadratic residues. So to honor him this tournament was named as Paley tournament. These results were extended by Bollobas for prime powers. Modular super edge trimagic labeling and modular super vertex magic total labeling has been investigated in this paper.

Complementary cycles in regular bipartite tournaments: a proof of Manoussakis, Song and Zhang Conjecture

Let D be a k-regular bipartite tournament. We show that, for every even p with 4≤p≤|V(D)|−4, D has a cycle C of size p such that D\\C is Hamiltonian unless D is isomorphic to a special digraph, F4k. This result proves a conjecture of Manoussakis, Song and Zhang.

Hypervolume Subset Selection in Two Dimensions: Formulations and Algorithms.

The hypervolume subset selection problem consists of finding a subset, with a given cardinality k, of a set of nondominated points that maximizes the hypervolume indicator. This problem arises in selection procedures of evolutionary algorithms for multiobjective optimization, for which practically efficient algorithms are required. In this article, two new formulations are provided for the two-dimensional variant of this problem. The first is a (linear) integer programming formulation that can be solved by solving its linear programming relaxation. The second formulation is a k-link shortest path formulation on a special digraph with the Monge property that can be solved by dynamic programming in [Formula: see text] time. This improves upon the result of [Formula: see text] in Bader ( 2009 ), and slightly improves upon the result of [Formula: see text] in Bringmann etal. ( 2014b ), which was developed independently from this work using different techniques. Numerical results are shown for several values of n and k.

Complementary cycles in regular bipartite tournaments

We show that every k-regular bipartite tournament B with k≥3 has two complementary cycles of lengths 6 and |V(B)|−6, unless B is isomorphic to a special digraph.

Parallel dynamical systems over special digraph classes

In a previous work, for parallel dynamical systems over digraphs corresponding to the simplest Boolean functions AND and OR, we proved that only fixed or eventually fixed points appear, as it occurs over undirected dependency graphs. Nevertheless, for general Boolean functions, it was shown that any period can appear, depending on the Boolean function that infers the global evolution operator of the system and on the structure of the dependency digraph. Motivated by these results, in this work, we analyse the orbit structure of parallel discrete dynamical systems over some special digraph classes.

Binary linear programming solutions and non-approximability for control problems in voting systems

In this paper we consider constructive control by adding and deleting candidates in Copeland and Llull voting systems from a theoretical and an experimental point of view.We show how to characterize the optimization versions of these four control problems as special digraph problems and binary linear programming formulations of linear size. Our digraph characterizations allow us to prove the hardness of approximations with absolute performance guarantee for optimal constructive control by deleting candidates in Copeland and by adding candidates in Llull voting schemes and the nonexistence of efficient approximation schemes for optimal constructive control by adding and deleting candidates in Copeland and Llull voting schemes. Our experimental study of running times using LP solvers shows that for a lot of practical instances even the hard control problems can be solved very efficiently.

Heuristic algorithm for finding the maximum independent set

A heuristic algorithm is developed for finding the maximum independent set of vertices in an undirected graph. To this end, the technique of finite partially ordered sets is used, in particular, the technique of partitioning such a set into a minimum number of chains. A special digraph is constructed and a solution algorithm is proposed on the basis of a hypothesis about its properties. Some experimental data are presented for well-known examples.

Crown-free highly arc-transitive digraphs

Abstract. We construct a family of infinite, non-locally finite highly arc-transitive digraphs which do not have universal reachability relation and which omit special digraphs called `crowns'. Moreover, there is no homomorphism from any of our digraphs onto Z $Z$ . The methods are adapted from [6] and [7].

The resound spelling of the coincidable diphthong [ie] in Lithuanian headlines and Ƶodrodyje toie kningelieie essontiu żodiù of Epitome Historiae Sacrae published by Simonas Daukantas in 1838

The usage of the special digraph type appears more various and frequent in Prasma (printed in 1837) than in Ƶodrodis (published a year later). This kind of digraph almost vanished from Daukantas’ dictionaries after 1838. Ƶodrodis is considerable to appear as a transitive work in which the type of characters such as and are used. Daukantas probably was translating Lithuanian headlines of the Publication Epitome Historiae Sacrae somewhat later, just at the same time when northern Samogitians decided to mark [ẹi] mainly by the symbol of the type.

On Hamiltonian Paths and a Chain Metric

A recurrent formula and an ordinary formula are proposed for the number of Hamiltonian paths on special digraphs generated by a chain metric on i>n-permutations.

Decomposition of Directed Graphs

A composition for directed graphs which generalizes the substitution (or X-join) composition of graphs and digraphs, as well as the graph version of set-family composition, is described. It is proved that a general decomposition theory can be applied to the resulting digraph decomposition. A consequence is a theorem which asserts the uniqueness of a decomposition of any digraph, each member of the decomposition being either indecomposable or “special”. The special digraphs are completely characterized; they are members of a few interesting classes. Efficient decomposition algorithms are also presented.

Cycles in bipartite tournaments

It is shown that if an oriented complete bipartite graph has a directed cycle of length 2 n, then it has directed cycles of all smaller even lengths unless n is even and the 2 n-cycle induces one special digraph.