Arc-disjoint Cycles Research Articles

Packing arc-disjoint cycles in oriented graphs

Arc-Disjoint Cycle Packing is a classical NP-complete problem and we study it from two perspectives: (1) by restricting the cycles in the packing to be of a fixed length, and (2) by restricting the inputs to bipartite tournaments. Focusing first on Arc-Disjointr-Cycle Packing (where the cycles in the packing are required to be of length r), we show NP-completeness in oriented graphs with girth r for each r≥3 and study the parameterized complexity of the problem with respect to two parameterizations (solution size and vertex cover size) for r=4 in oriented graphs. Moving on to Arc-Disjoint Cycle Packing in bipartite tournaments, we show that every bipartite tournament either contains k arc-disjoint cycles or has a feedback arc set of size at most 7(k−1). This result adds to the set of Erdös-Pósa-type results known in the combinatorics literature for packing and covering problems.

Packing Arc-Disjoint Cycles in Tournaments

A tournament is a directed graph in which there is a single arc between every pair of distinct vertices. Given a tournament T on n vertices, we explore the classical and parameterized complexity of the problems of determining if T has a cycle packing (a set of pairwise arc-disjoint cycles) of size k and a triangle packing (a set of pairwise arc-disjoint triangles) of size k. We refer to these problems as Arc-disjoint Cycles in Tournaments (ACT) and Arc-disjoint Triangles in Tournaments (ATT), respectively. Although the maximization version of ACT can be seen as the dual of the well-studied problem of finding a minimum feedback arc set (a set of arcs whose deletion results in an acyclic graph) in tournaments, surprisingly no algorithmic results seem to exist for ACT. We first show that ACT and ATT are both NP-complete. Then, we show that the problem of determining if a tournament has a cycle packing and a feedback arc set of the same size is NP-complete. Next, we prove that ACT is fixed-parameter tractable via a $$2^{\mathcal {O}(k \log k)} n^{\mathcal {O}(1)}$$ -time algorithm and admits a kernel with $$\mathcal {O}(k)$$ vertices. Then, we show that ATT too has a kernel with $$\mathcal {O}(k)$$ vertices and can be solved in $$2^{\mathcal {O}(k)} n^{\mathcal {O}(1)}$$ time. Afterwards, we describe polynomial-time algorithms for ACT and ATT when the input tournament has a feedback arc set that is a matching. We also prove that ACT and ATT cannot be solved in $$2^{o(\sqrt{n})} n^{\mathcal {O}(1)}$$ time under the exponential-time hypothesis.

A note on arc-disjoint cycles in tournaments

A note on arc-disjoint cycles in tournaments

Arc-Disjoint and Edge-Disjoint Hamilton Cycles in Circulants with Two Jumps

In this paper we give new necessary and sufficient conditions for a directed circulant with vertices of outdegree two to have a pair of arc-disjoint Hamilton cycles. These conditions explicitly identify a pair of arc-disjoint Hamilton cycles if such cycles exist. In addition, we give necessary and sufficient conditions for an undirected circulant with vertices of degree four to have a specific pair of edge-disjoint Hamilton cycles.

All that glisters is not galled

Galled trees, evolutionary networks with isolated reticulation cycles, have appeared under several slightly different definitions in the literature. In this paper, we establish the actual relationships between the main four such alternative definitions: namely, the original galled trees, level-1 networks, nested networks with nesting depth 1, and evolutionary networks with arc-disjoint reticulation cycles.

Multidimensional Manhattan Street Networks

We formally define the $n$-dimensional Manhattan street network $M_n$—a special case of an $n$-regular digraph—and we study some of its structural properties. In particular, we show that $M_n$ is a Cayley digraph, which can be seen as a subgroup of the $n$-dimensional version of the wallpaper group $pgg$. These results induce a useful new representation of $M_n$, which can be applied to design a local (shortest-path) routing algorithm and to study some other metric properties, such as the diameter. We also show that the $n$-dimensional Manhattan street networks are Hamiltonian and, in the standard case (that is, in dimension two), we give sufficient conditions for a $2$-dimensional Manhattan street network to be decomposable into two arc-disjoint Hamiltonian cycles.

Disjoint Cycles in Eulerian Digraphs and the Diameter of Interchange Graphs

Let R=(r1, …, rm) and S=(s1, …, sn) be nonnegative integral vectors with ∑ri=∑sj. Let A(R, S) denote the set of all m×n {0, 1}-matrices with row sum vector R and column sum vector S. Suppose A(R, S)≠∅. The interchange graphG(R, S) of A(R, S) was defined by Brualdi in 1980. It is the graph with all matrices in A(R, S) as its vertices and two matrices are adjacent provided they differ by an interchange matrix. Brualdi conjectured that the diameter of G(R, S) cannot exceed mn/4. A digraph G=(V, E) is called Eulerian if, for each vertex u∈V, the outdegree and indegree of u are equal. We first prove that any bipartite Eulerian digraph with vertex partition sizes m, n, and with more than (17−1)mn/4 (≈0.78mn) arcs contains a cycle of length at most 4. As an application of this, we show that the diameter of G(R, S) cannot exceed (3+17)mn/16 (≈0.445mn). The latter result improves a recent upper bound on the diameter of G(R, S) by Qian. Finally, we present some open problems concerning the girth and the maximum number of arc-disjoint cycles in an Eulerian digraph.

Tournaments as Feedback Arc Sets

We examine the size $s(n)$ of a smallest tournament having the arcs of an acyclic tournament on $n$ vertices as a minimum feedback arc set. Using an integer linear programming formulation we obtain lower bounds $s(n) \geq 3n - 2 - \lfloor \log_2 n \rfloor$ or $s(n) \geq 3n - 1 - \lfloor \log_2 n \rfloor$, depending on the binary expansion of $n$. When $n = 2^k - 2^t$ we show that the bounds are tight with $s(n) = 3n - 2 - \lfloor \log_2 n \rfloor$. One view of this problem is that if the 'teams' in a tournament are ranked to minimize inconsistencies there is some tournament with $s(n)$ 'teams' in which $n$ are ranked wrong. We will also pose some questions about conditions on feedback arc sets, motivated by our proofs, which ensure equality between the maximum number of arc disjoint cycles and the minimum size of a feedback arc set in a tournament.

Compositions of Graphs and Polyhedra IV: Acyclic Spanning Subgraphs

Given a directed graph D that has a two-vertex cut, this paper describes a technique to derive a linear system that defines the acyclic subgraph polytope of D from systems related to the pieces. It also gives a technique to describe facets of this polytope by composition of facets for the pieces. The authors prove that, if the systems for the pieces are totally dual integral (TDI), then the system for D is also. The authors prove that the “cycle inequalities” form a TDI system for any orientation of $K_5$. These results are combined with Lucchesi–Younger theorem and a theorem of Wagner to prove that, for graphs with no $K_{3,3} $ minor, the cycle inequalities characterize the acyclic subgraph polytope and form a TDI system. This shows that, for this class of graphs, the cardinality of a minimum feedback set is equal to the maximum number of arc disjoint cycles. For planar graphs, this is a consequence of the Lucchesi–Younger theorem.

A Minimax Arc Theorem for Reducible Flow Graphs

A conjecture of Frank and Gyarfas is established by proving that the cardinality of a minimum feedback arc set in a reducible flow graph is equal to the cardinality of a maximum collection of arc disjoint cycles.

An algorithm for generation of s‐t acyclic graphs

AbstractThe problem of finding all s‐t acyclic graphs between two given vertices of a digraph is interesting in relation to the evaluation of the reliability of a communication network. There is already an algorithm which generates all s‐t acyclic digraphs without duplication by opening an arc or a set of arcs one by one starting with a given graph. But the efficiency of that algorithm decreases when the number of cycles in the graph becomes large. This paper proposes an algorithm which generates s‐t acyclic graphs without duplication by combining s‐t paths obtained by a depth‐first search. Compared with the existing algorithms (i) the execution time of this algorithm is not affected much by the number of arc disjoint cycles of the graph; (ii) when the generation is stopped at an arbitrary level i, all the s‐t acyclic graphs composed of paths of length i or less will have been generated. Thus they can be used to approximate the reliability.

Permutation-partition pairs. II. Bounds on the genus of the amalgamation of graphs

Bounds are derived on the extent to which the parameter μ ( P , ∏ ) \mu (P,\,\prod ) can fail to be additive over disjoint permutations. This is done by associating an Eulerian digraph to each such pair and relating the maximum orbiticity μ ( P , ∏ ) \mu (P,\,\prod ) to the decompositions of this digraph’s arc set into arc disjoint cycles. These bounds are then applied to obtain information about the genus of the amalgamation of graphs.