Symmetric Digraph Research Articles

The λ-Fold Spectrum Problem for the Orientations of the Eight-Cycle

A D-decomposition of a graph (or digraph) G is a partition of the edge set (or arc set) of G into subsets, where each subset induces a copy of the fixed graph D. Graph decomposition finds motivation in numerous practical applications, particularly in the realm of symmetric graphs, where these decompositions illuminate intricate symmetrical patterns within the graph, aiding in various fields such as network design, and combinatorial mathematics, among various others. Of particular interest is the case where G is K*λKv*, the λ-fold complete symmetric digraph on v vertices, that is, the digraph with λ directed edges in each direction between each pair of vertices. For a given digraph D, the set of all values v for which K*λKv* has a D-decomposition is called the λ-fold spectrum of D. An eight-cycle has 22 non-isomorphic orientations. The λ-fold spectrum problem has been solved for one of these oriented cycles. In this paper, we provide a complete solution to the λ-fold spectrum problem for each of the remaining 21 orientations.

Deterministic n-person shortest path and terminal games on symmetric digraphs have Nash equilibria in pure stationary strategies

We prove that a deterministic n-person shortest path game has a Nash equlibrium in pure and stationary strategies if it is edge-symmetric (that is (u, v) is a move whenever (v, u) is, apart from moves entering terminal vertices) and the length of every move is positive for each player. Both conditions are essential, though it remains an open problem whether there exists a NE-free 2-person non-edge-symmetric game with positive lengths. We provide examples for NE-free 2-person edge-symmetric games that are not positive. We also consider the special case of terminal games (shortest path games in which only terminal moves have nonzero length, possibly negative) and prove that edge-symmetric n-person terminal games always have Nash equilibria in pure and stationary strategies. Furthermore, we prove that an edge-symmetric 2-person terminal game has a uniform (subgame perfect) Nash equilibrium, provided any infinite play is worse than any of the terminals for both players.

Completing the solution of the directed Oberwolfach problem with cycles of equal length

AbstractIn this paper, we give a solution to the last outstanding case of the directed Oberwolfach problem with tables of uniform length. Namely, we address the two‐table case with tables of equal odd length. We prove that the complete symmetric digraph on vertices, denoted , admits a resolvable decomposition into directed cycles of odd length . This completely settles the directed Oberwolfach problem with tables of uniform length.

Upper Bounds of the Generalized Competition Indices of Symmetric Primitive Digraphs with d Loops

A digraph (D) is symmetric if (u,v) is an arc of D and if (v,u) is also an arc of D. If a symmetric digraph is primitive and contains d loops, then it is said to be a symmetric primitive digraph with d loops. The m-competition index (generalized competition index) of a digraph is an extension of the exponent and the scrambling index. The m-competition index has been applied to memoryless communication systems in recent years. In this article, we assume that Sn(d) represents the set of all symmetric primitive digraphs of n vertices with d loops, where 1≤d≤n. We study the m-competition indices of Sn(d) and give their upper bounds, where 1≤m≤n. Furthermore, for any integer m satisfying 1≤m≤n, we find that the upper bounds of the m-competition indices of Sn(d) can be reached.

How to construct the symmetric cycle of length 5 using Haj\'os construction with an adapted Rank Genetic Algorithm

In 2020 Bang-Jensen et. al. generalized the Haj\'os join of two graphs to the class of digraphs and generalized several results for vertex colorings in digraphs. Although, as a consequence of these results, a digraph can be obtained by Haj\'os constructions (directed Haj\'os join and identifying non-adjacent vertices), determining the Haj\'os constructions to obtain the digraph is a complex problem. In particular, Bang-Jensen et al. posed the problem of determining the Haj\'os operations to construct the symmetric 5-cycle from the complete symmetric digraph of order 3 using only Haj\'os constructions. We successfully adapted a rank-based genetic algorithm to solve this problem by the introduction of innovative recombination and mutation operators from graph theory. The Haj\'os Join became the recombination operator and the identification of independent vertices became the mutation operator. In this way, we were able to obtain a sequence of only 16 Haj\'os operations to construct the symmetric cycle of order 5.

A Study on Network Graph-PW, Network Symmetric Digraph-PW, Change Network Graph-PW and Change Network Symmetric Digraph- PW

Networks play an important role in electrical and electronic engineering. It depends on what area of electrical and electronic engineering, for example, there is a lot more abstract mathematics in communication theory and signal processing and networking, etc. Networks involve nodes communicating with each other. Graph theory has found considerable use in this area. In this paper, we introduce some new Networks such as Graph-PW, Network Symmetric Digraph-PW, Change Network Graph-PW, and Change Network Symmetric Digraph- PW. Moreover, several theorems and results of these networks have been studied.

Directed Tree Connectivity of Symmetric Digraphs and Complete Bipartite Digraphs

Sun and Yeo introduced the concept of directed tree connectivity, including the generalized [Formula: see text]-vertex-strong connectivity, [Formula: see text] and generalized [Formula: see text]-arc-strong connectivity, [Formula: see text] [Formula: see text], which could be seen as a generalization of classical connectivity of digraphs and a natural extension of the well-established undirected tree connectivity. In this paper, we study the directed tree connectivity of symmetric digraphs and complete bipartite digraphs. We give lower bounds for the two parameters [Formula: see text] and [Formula: see text] on symmetric digraphs. We also determine the precise values of [Formula: see text] for every [Formula: see text] and [Formula: see text] for [Formula: see text], where [Formula: see text] is a complete bipartite digraph of order [Formula: see text].

Distinguishing arc-colourings of symmetric digraphs

A symmetric digraph is obtained from an undirected graph by replacing each edge uv of by a pair of opposite arcs and . An arc-colouring of a digraph is called distinguishing if the only automorphism preserving it is the identity. The least number of colours in a distinguishing arc-colouring, not necessarily proper, of is called the distinguishing index D'(). We study bounds for D'(). For proper distinguishing arc-colourings, the least number of colours is called the distinguishing chromatic index of . There are 15 possible types of proper arc-colourings of a digraph depending on the definition of adjacent arcs. In this paper we investigate distinguishing chromatic indices of for the nine remaining types not considered in our two previous papers. We formulate several conjectures.

Packing strong subgraph in digraphs

In this paper, we study two types of strong subgraph packing problems in digraphs, including internally disjoint strong subgraph packing problem and arc-disjoint strong subgraph packing problem. These problems can be viewed as generalizations of the famous Steiner tree packing problem and are closely related to the strong arc decomposition problem. We first prove the NP-completeness for the internally disjoint strong subgraph packing problem restricted to symmetric digraphs and Eulerian digraphs. Then we get inapproximability results for the arc-disjoint strong subgraph packing problem and the internally disjoint strong subgraph packing problem. Finally we study the arc-disjoint strong subgraph packing problem restricted to digraph compositions and obtain some algorithmic results by utilizing the structural properties.

Directed Steiner tree packing and directed tree connectivity

AbstractFor a digraph , and a set with and , an ‐tree is an out‐tree rooted at with . Two ‐trees and are said to be arc‐disjoint if . Two arc‐disjoint ‐trees and are said to be internally disjoint if . Let and be the maximum number of internally disjoint and arc‐disjoint ‐trees in , respectively. The generalized ‐vertex‐strong connectivity of is defined as Similarly, the generalized ‐arc‐strong connectivity of is defined as The generalized ‐vertex‐strong connectivity and generalized ‐arc‐strong connectivity are also called directed tree connectivity which extends the well‐established tree connectivity on undirected graphs to directed graphs and could be seen as a generalization of classical connectivity of digraphs. In this paper, we completely determine the complexity for both and on general digraphs, symmetric digraphs, and Eulerian digraphs. In particular, among our results, we prove and use the NP‐completeness of 2‐linkage problem restricted to Eulerian digraphs. We also give sharp bounds and equalities for the two parameters and .

χ-Diperfect digraphs

Let D be a digraph. A coloring C and a path P of D are orthogonal if P contains exactly one vertex of each color class in C. In 1982, Berge defined the class of χ-diperfect digraphs. A digraph D is χ-diperfect if for every minimum coloring C of D, there exists a path P orthogonal to C and this property holds for every induced subdigraph of D. Berge showed that every symmetric digraph is χ-diperfect, as well as every digraph whose underlying graph is perfect. However, he also showed that not every super-orientation of an odd cycle or complement of an odd cycle is χ-diperfect. Non-χ-diperfect super-orientations of odd cycles and their complements may play an important role in the characterization of χ-diperfect digraphs, similarly to the role they play in the characterization of perfect graphs. In this paper, we present a characterization of super-orientations of odd cycles and a characterization of super-orientations of complements of odd cycles that are χ-diperfect. Moreover, we show that certain classes of digraphs that are free of such non-χ-diperfect structures are χ-diperfect.

Proper distinguishing arc-colourings of symmetric digraphs

A symmetric digraph G↔ arises from a simple graph G by substituting each edge uv by a pair of opposite arcs uv→,vu→. An arc-colouring c of G↔ is distinguishing if the only automorphism of G↔ preserving c is the identity. We study four types of proper arc-colourings of G↔ corresponding to four definitions of adjacency of arcs. For each type, we investigate the distinguishing chromatic index of G↔, i.e. the least number of colours in a distinguishing proper colouring of G↔. We also determine tight bounds for chromatic indices of G↔, i.e. for the least numbers of colours in each type of proper colourings. Colourings of arcs of a symmetric digraph G↔ are equivalent to colourings of halfedges of the graph G, which have applications in computer science.

Trail-Connected Digraphs with Given Local Structures

For a digraph [Formula: see text], if [Formula: see text] contains a spanning closed trail, then [Formula: see text] is supereulerian. If for any pairs vertices [Formula: see text] and [Formula: see text] of [Formula: see text], [Formula: see text] contains both a spanning [Formula: see text]-trail and a spanning [Formula: see text]-trail, then [Formula: see text] is strongly trail-connected. If [Formula: see text] is a strongly trail-connected digraph, then [Formula: see text] is a supereulerian digraph. Algefari et al. proved that every symmetrically connected digraph and every partially symmetric digraph are supereulerian. In this paper, we prove that every symmetrically connected digraph and every partially symmetric digraph are strongly trail-connected digraphs.

The linear $ k $-arboricity of digraphs

<abstract><p>A linear $ k $-diforest is a directed forest in which every connected component is a directed path of length at most $ k $. The linear $ k $-arboricity of a digraph $ D $ is the minimum number of linear $ k $-diforests needed to partition the arcs of $ D $. In this paper, we study the linear $ k $-arboricity for digraphs, and determine the linear $ 3 $-arboricity and linear $ 2 $-arboricity for symmetric complete digraphs and symmetric complete bipartite digraphs.</p></abstract>

Balanced flows for transshipment problems

A transshipment problem (G,d,λ) is modeled by a directed graph G=(V,E) with weighted vertices (nodes) d=(dv∣v∈V) and weighted directed edges (arcs) λ=(λe∣e∈E) interpreted as follows: G is a communication or transportation network, for example, a pipeline; each arc e∈E is a one-way communication line, road or pipe of capacity λe, while every vertex v∈V is a node of production dv>0, consumption dv<0, or just transition dv=0. A non-negative flow x=(xe∣e∈E) is called weakly feasible if for each v∈V the algebraic sum of flows, over all directed edges incident to v, equals dv, or shorter, if AGx=d, where AG is the vertex-edge incidence matrix of G. A weakly feasible flow x is called feasible if xe≤λe for all e∈E. We consider weakly feasible but not necessarily feasible flows, that is, inequalities xe>λe are allowed. However, such an excess is viewed as unwanted (dangerous) and so we minimize the excess ratio vector r=(re=xe/λe∣e∈E) lexicographically. More precisely, first, we look for weakly feasible flows minimizing the maximum of re over all e∈E; among all such flows we look for those that minimize the second largest coordinate of r, etc. Clearly, in |E| such steps we obtain a unique lexmin vector r. We show that the corresponding balanced flow is also unique and represents the lexmin solution for problem (G,d,λ). Using the Gale–Hoffman inequalities, we construct it in polynomial time, provided vectors d and λ are rational.For symmetric digraphs the problem was solved by Gurvich and Gvishiani in 1984. Here we extend this result to arbitrary directed graphs. Furthermore, we simplify the algorithm and proofs applying the classic criterion of existence of a feasible flow for (G,d,λ) obtained by Gale and Hoffman in late 1950s.

Path and Star Decomposition of Knodel and Fibonacci Digraphs

In broadcasting and gossiping, communication structures are modelled using graphs. In simplex model, a communication link sends messages in a particular direction. The simplex network is modelled by directed graphs. Certain Knodel graphs are optimal broadcast graphs. This article defines the two layer representation of Knodel digraphs and Fibonacci digraphs-both are symmetric regular bipartite digraphs. A Knodel digraph is a symmetric digraph produced by replacing each edge of an undirected Knodel graph by a symmetric pair of directed edges. A Fibonacci digraph is a symmetric digraph produced by replacing each edge of an undirected Fibonacci graph by a symmetric pair of directed edges. Decomposition of a given graph is the partitioning of edge set so that the given graph is split in to sub-structures. If all such sub-structures are isomorphic, the decomposition is isomorphic decomposition. The isomorphic decomposition of the Knodel digraphs and Fibonacci digraphs in to isomorphic paths and stars is presented. The star decomposition of both Knodel digraph and Fibonacci digraph leads to the decomposition of equi-bipartite complete digraphs in to two classes of stars.

On the Complexity of Broadcast Domination and Multipacking in Digraphs

We study the complexity of the two dual covering and packing distance-based problems Broadcast Domination and Multipacking in digraphs. A dominating broadcast of a digraph D is a function \(f:V(D)\rightarrow {\mathbb {N}}\) such that for each vertex v of D, there exists a vertex t with \(f(t)>0\) having a directed path to v of length at most f(t). The cost of f is the sum of f(v) over all vertices v. A multipacking is a set S of vertices of D such that for each vertex v of D and for every integer d, there are at most d vertices from S within directed distance at most d from v. The maximum size of a multipacking of D is a lower bound to the minimum cost of a dominating broadcast of D. Let Broadcast Domination denote the problem of deciding whether a given digraph D has a dominating broadcast of cost at most k, and Multipacking the problem of deciding whether D has a multipacking of size at least k. It is known that Broadcast Domination is polynomial-time solvable for the class of all undirected graphs (that is, symmetric digraphs), while polynomial-time algorithms for Multipacking are known only for a few classes of undirected graphs. We prove that Broadcast Domination and Multipacking are both NP-complete for digraphs, even for planar layered acyclic digraphs of small maximum degree. Moreover, when parameterized by the solution cost/solution size, we show that the problems are respectively W[2]-hard and W[1]-hard. We also show that Broadcast Domination is FPT on acyclic digraphs, and that it does not admit a polynomial kernel for such inputs, unless the polynomial hierarchy collapses to its third level. In addition, we show that both problems are FPT when parameterized by the solution cost/solution size together with the maximum (out-)degree, and as well, by the vertex cover number. Finally, we give for both problems polynomial-time algorithms for some subclasses of acyclic digraphs.

Chordality of locally semicomplete and weakly quasi-transitive digraphs

Chordal graphs are important in the structural and algorithmic graph theory. A digraph analogue of chordal graphs was introduced by Haskin and Rose in 1973 but has not been the subject of active studies until recently when a characterization of semicomplete chordal digraphs in terms of forbidden subdigraphs was found by Meister and Telle.Locally semicomplete digraphs, quasi-transitive digraphs, and extended semicomplete digraphs are amongst the most popular generalizations of semicomplete digraphs. We extend the forbidden subdigraph characterization of semicomplete chordal digraphs to locally semicomplete chordal digraphs. We introduce a new class of digraphs, called weakly quasi-transitive digraphs, which contains quasi-transitive digraphs, symmetric digraphs, and extended semicomplete digraphs, but is incomparable to the class of locally semicomplete digraphs. We show that weakly quasi-transitive digraphs can be recursively constructed by simple substitutions from transitive oriented graphs, semicomplete digraphs, and symmetric digraphs. This recursive construction of weakly quasi-transitive digraphs, similar to the one for quasi-transitive digraphs, demonstrates the naturalness of the new digraph class. As a by-product, we prove that the forbidden subdigraphs for semicomplete chordal digraphs are the same for weakly quasi-transitive chordal digraphs. The forbidden subdigraph characterization of weakly quasi-transitive chordal digraphs generalizes not only the recent results on quasi-transitive chordal digraphs and extended semicomplete chordal digraphs but also the classical results on chordal graphs.

On decomposing the complete symmetric digraph into orientations of K_4-e

On decomposing the complete symmetric digraph into orientations of K_4-e

Colouring of generalized signed triangle-free planar graphs

We view an undirected graph G as a symmetric digraph, where each edge xy is replaced by two opposite arcs e=(x,y) and e−1=(y,x). Assume S is an inverse closed subset of permutations of positive integers. We say G is S-k-colourable if for any mapping σ:E(G)→S with σ(x,y)=(σ(y,x))−1, there is a mapping f:V(G)→[k]={1,2,…,k} such that σe(f(x))≠f(y) for each arc e=(x,y). The concept of S-k-colourable is a common generalization of several other colouring concepts. This paper is focused on finding the sets S such that every triangle-free planar graph is S-3-colourable. Such a set S is called TFP-good. Grötzsch’s theorem is equivalent to say that S={id} is TFP-good. We prove that for any inverse closed subset S of S3 which is not isomorphic to {id,(12)}, S is TFP-good if and only if either S={id} or there exists a∈[3] such that for each π∈S, π(a)≠a. It remains an open question to determine whether or not S={id,(12)} is TFP-good.