Semicomplete Digraphs Research Articles

Strong arc decompositions of split digraphs

AbstractA strong arc decomposition of a digraph is a partition of its arc set into two sets such that the digraph is strong for . Bang‐Jensen and Yeo conjectured that there is some such that every ‐arc‐strong digraph has a strong arc decomposition. They also proved that with one exception on four vertices every 2‐arc‐strong semicomplete digraph has a strong arc decomposition. Bang‐Jensen and Huang extended this result to locally semicomplete digraphs by proving that every 2‐arc‐strong locally semicomplete digraph which is not the square of an even cycle has a strong arc decomposition. This implies that every 3‐arc‐strong locally semicomplete digraph has a strong arc decomposition. A split digraph is a digraph whose underlying undirected graph is a split graph, meaning that its vertices can be partitioned into a clique and an independent set. Equivalently, a split digraph is any digraph which can be obtained from a semicomplete digraph by adding a new set of vertices and some arcs between and . In this paper, we prove that every 3‐arc‐strong split digraph has a strong arc decomposition which can be found in polynomial time and we provide infinite classes of 2‐strong split digraphs with no strong arc decomposition. We also pose a number of open problems on split digraphs.

Arc-disjoint out- and in-branchings in compositions of digraphs

An out-branching Bu+ (in-branching Bu−) in a digraph D is a connected spanning subdigraph of D in which every vertex except the vertex u, called the root, has in-degree (out-degree) one. A good(u,v)-pair in D is a pair of branchings Bu+,Bv− which have no arc in common. Thomassen proved that it is NP-complete to decide if a digraph has any good pair. A digraph is semicomplete if it has no pair of non-adjacent vertices. A semicomplete composition is any digraph D which is obtained from a semicomplete digraph S by substituting an arbitrary digraph Hx for each vertex x of S.Recently the authors of this paper gave a complete classification of semicomplete digraphs which have a good (u,v)-pair, where u,v are prescribed vertices. They also gave a polynomial algorithm which for a given semicomplete digraph D and vertices u,v of D, either produces a good (u,v)-pair in D or a certificate that D has no such pair. In this paper we show how to use the result for semicomplete digraphs to completely solve the problem of characterizing semicomplete compositions which have a good (u,v)-pair for given vertices u,v. Our solution implies that the problem of deciding the existence of a good (u,v)-pair and finding such a pair when it exists is polynomially solvable for all semicomplete compositions. We also completely solve the problem of deciding the existence of a good (u,v)-pair and finding one when it exists for digraphs that are compositions of transitive digraphs. Combining these two results we obtain a polynomial algorithm for deciding whether a given quasi-transitive digraph D has a good (u,v)-pair for given vertices u,v of D. This proves a conjecture of Bang-Jensen and Gutin from 1998.

Arc‐disjoint out‐branchings and in‐branchings in semicomplete digraphs

AbstractAn out‐branching (in‐branching ) in a digraph is a connected spanning subdigraph of in which every vertex except the vertex , called the root, has in‐degree (out‐degree) one. It is well known that there exists a polynomial algorithm for deciding whether a given digraph has arc‐disjoint out‐branchings with prescribed roots ( is part of the input). In sharp contrast to this, it is already NP‐complete to decide if a digraph has one out‐branching which is arc‐disjoint from some in‐branching. A digraph is semicomplete if it has no pair of nonadjacent vertices. A tournament is a semicomplete digraph without directed cycles of length 2. In this paper we give a complete classification of semicomplete digraphs that have an out‐branching which is arc‐disjoint from some in‐branching where are prescribed vertices of . Our characterization, which is surprisingly simple, generalizes a complicated characterization for tournaments from 1991 by the first author and our proof implies the existence of a polynomial algorithm for checking whether a given semicomplete digraph has such a pair of branchings for prescribed vertices and construct a solution if one exists. This confirms a conjecture of Bang‐Jensen for the case of semicomplete digraphs.

Comparability digraphs: An analogue of comparability graphs

Comparability graphs are a popular class of graphs. We introduce as the digraph analogue of comparability graphs the class of comparability digraphs. We show that many concepts such as implication classes and the knotting graph for a comparability graph can be naturally extended to a comparability digraph. We give a characterization of comparability digraphs in terms of their knotting graphs.Semicomplete comparability digraphs contain in a sense all comparability graphs. One instrumental technique for analyzing the structure of comparability graphs is the Triangle Lemma for graphs (cf. Golumbic (1980) [7]). Using Triangle Lemma, Golumbic proved that a graph is a comparability graph if and only if no implication class of the graph is equal to its inverse. We give examples to show that a similar statement does not hold for general digraphs. We prove that it holds for semicomplete digraphs, that is, a semicomplete digraph is a comparability digraph if and only if no implication class of the digraph is equal to its inverse. The proof is done by generalizing the Triangle Lemma for graphs to semicomplete digraphs. This yields an O(n3)-time recognition algorithm for semicomplete comparability digraphs where n is the number of vertices of the input digraph. The correctness of the algorithm also implies several other characterizations for semicomplete comparability digraphs, akin to those for comparability graphs.

Spanning 3-strong tournaments in 5-strong semicomplete digraphs

A digraph is strong if it has a directed path from x to y for every ordered pair of distinct vertices x,y and it is k-strong if it has at least k+1 vertices and remains strong when we delete any set of at most k−1 vertices. A digraph D is a tournament (semicomplete digraph) if there is precisely (at least) one arc between any pair of distinct vertices of D. A subdigraph of a digraph D is called spanning if it contains all vertices of D. In this paper, we show that every 5-strong semicomplete digraph on at least 9 vertices contains a spanning 3-strong tournament. This shows that the conjecture of Bang-Jensen and Jordán [Discrete Mathematics 310 (2010) 1424–1428] holds for k=3.

Semicomplete compositions of digraphs

Let T be a digraph with vertices u1,…,ut (t≥2), and let H1,…,Ht be digraphs such that Hi has vertices ui,ji, where 1≤ji≤ni. The composition Q=T[H1,…,Ht] is a digraph with vertex setV(Q)={ui,ji:1≤i≤t,1≤ji≤ni} and arc set(⋃i=1tA(Hi))⋃(⋃uiup∈A(T){ui,jiup,qp:1≤ji≤ni,1≤qp≤np}). The composition Q=T[H1,…,Ht] is a semicomplete composition if T is semicomplete, i.e., there is at least one arc between every pair of vertices. Digraph compositions generalize some families of digraphs, including quasi-transitive digraphs, extended semicomplete digraphs and lexicographic product digraphs. In particular, strong semicomplete compositions form a significant generalization of strong quasi-transitive digraphs.In this paper, we study the structural properties of semicomplete compositions and obtain results on connectivity, paths, cycles and strong spanning subdigraphs. Our results show that this class of digraphs shares some nice properties of quasi-transitive digraphs.

A minimum semi-degree sufficient condition for one-to-many disjoint path covers in semicomplete digraphs

Let D be a digraph. We define the minimum semi-degree of D as δ0(D):=min⁡{δ+(D),δ−(D)}. Let k be a positive integer, and let S={s} and T={t1,…,tk} be any two disjoint subsets of V(D). A set of k internally disjoint paths joining source set S and sink set T that cover all vertices of D are called a one-to-many k-disjoint directed path cover (k-DDPC for short) of D. A digraph D is semicomplete if for every pair x,y of vertices of it, there is at least one arc between x and y.In this paper, we prove that every semicomplete digraph D of sufficiently large order n with δ0(D)≥⌈(n+k−1)/2⌉ has a one-to-many k-DDPC joining any disjoint source set S and sink set T, where S={s},T={t1,…,tk}.

Improved results on linkage problems

A digraph D is k-linked if |D|≥2k, and whenever x1,…,xk,y1,…,yk are 2k distinct vertices of D, there exist vertex-disjoint paths P1,…,Pk such that Pi is a path from xi to yi for each i∈[k]. A digraph is strongly connected if it has a directed path from x to y for every ordered pair of distinct vertices x,y and it is strongly k-connected (briefly k-connected) if it has at least k+1 vertices and remains strongly connected when we delete any set of at most k−1 vertices. In this paper, we prove that every (13k−6)-connected semicomplete digraph with minimum out-degree at least 19k−6 is k-linked. This result extends the result of Bang-Jensen and Johansen (2022) [3] to semicomplete digraphs. Moreover, we show the following holds. Let n,k,l∈N, and let D be a digraph of order n. Suppose that D is (39k+9l−28)-connected with minimum degree at least n−l and minimum out-degree at least 46k+15l−35. Then D is k-linked.

Counting lines in semi-complete digraphs *

A digraph D= (V, A) is semi-complete if for each pair of distinct vertices x and y in V, either xy or yx belong to A. A subset ℓ of vertices is a line of D if there are two distinct vertices x and y such that for any vertex zε V, zε ℓ if and only if a directed shortest path exists containing x,y and z.A classic result proved by Erdös says that any set of n points in the Euclidean plane endowed with the Euclidean distance defines a metric space with at least n different lines unless there is a line containing the n points. Chen and Chvátal in 2008 conjectured that the same results is true for any metric spaces where lines are defined in a manner similar to above.In this paper we prove that in any semi-complete digraphs with n vertices the number of lines defined by vertices connected by an arc is at least n. Then, the quasi-metric spaces defined by semi-complete digraphs fulfill Chen and Chvátal conjecture in a stronger manner as, on the one hand, they always have at least n lines, and on the other hand, these n lines are defined by vertices at distance one.

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.

Characterizations and Directed Path-Width of Sequence Digraphs

Computing the directed path-width of a directed graph is an NP-hard problem. Even for digraphs of maximum semi-degree 3 the problem remains hard. We propose a decomposition of an input digraph G = (V,A) by a number k of sequences with entries from V, such that (u,v) ∈ A if and only if in one of the sequences there is an occurrence of u appearing before an occurrence of v. We present several graph theoretical properties of these digraphs. Among these we give forbidden subdigraphs of digraphs which can be defined by k = 1 sequence, which is a subclass of semicomplete digraphs. Given the decomposition of digraph G, we show an algorithm which computes the directed path-width of G in time mathcal {O}(kcdot (1+N)^{k}), where N denotes the maximum sequence length. This leads to an XP-algorithm w.r.t. k for the directed path-width problem. Our result improves the algorithms of Kitsunai et al. for digraphs of large directed path-width which can be decomposed by a small number of sequences and confirm their conjecture that semicompleteness is a useful restriction when considering digraphs.

Making a tournament k $k$‐strong

A digraph is k ${\bf{k}}$ -strong if it has n ≥ k + 1 $n\ge k+1$ vertices and every induced subdigraph on at least n − k + 1 $n-k+1$ vertices is strongly connected. A tournament is a digraph with no pair of nonadjacent vertices. We prove that every tournament on n ≥ k + 1 $n\ge k+1$ vertices can be made k $k$ -strong by adding no more than k + 1 2 $\left(\genfrac{}{}{0ex}{}{k+1}{2}\right)$ arcs. This solves a conjecture from 1994. A digraph is semicomplete if there is at least one arc between any pair of distinct vertices x , y $x,y$ . Since every semicomplete digraph contains a spanning tournament, the result above also holds for semicomplete digraphs. Our result also implies that for every k ≥ 2 $k\ge 2$ , every semicomplete digraph on at least 3 k − 1 $3k-1$ vertices can be made k $k$ -strong by reversing no more than k + 1 2 $\left(\genfrac{}{}{0ex}{}{k+1}{2}\right)$ arcs.

Spanning eulerian subdigraphs in semicomplete digraphs

AbstractA digraph is eulerian if it is connected and every vertex has its in‐degree equal to its out‐degree. Having a spanning eulerian subdigraph is thus a weakening of having a hamiltonian cycle. In this paper, we first characterize the pairs of a semicomplete digraph and an arc such that has a spanning eulerian subdigraph containing . In particular, we show that if is 2‐arc‐strong, then every arc is contained in a spanning eulerian subdigraph. We then characterize the pairs of a semicomplete digraph and an arc such that has a spanning eulerian subdigraph avoiding . In particular, we prove that every 2‐arc‐strong semicomplete digraph has a spanning eulerian subdigraph avoiding any prescribed arc. We also prove the existence of a (minimum) function such that every ‐arc‐strong semicomplete digraph contains a spanning eulerian subdigraph avoiding any prescribed set of arcs. We conjecture that and establish this conjecture for and when the arcs that we delete form a forest of stars. A digraph is eulerian‐connected if for any two distinct vertices , the digraph has a spanning ‐trail. We prove that every 2‐arc‐strong semicomplete digraph is eulerian‐connected. All our results may be seen as arc analogues of well‐known results on hamiltonian paths and cycles in semicomplete digraphs.

Towards the Small Quasi-Kernel Conjecture

Let $D=(V,A)$ be a digraph. A vertex set $K\subseteq V$ is a quasi-kernel of $D$ if $K$ is an independent set in $D$ and for every vertex $v\in V\setminus K$, $v$ is at most distance 2 from $K$. In 1974, Chvátal and Lovász proved that every digraph has a quasi-kernel. P. L. Erdős and L. A. Székely in 1976 conjectured that if every vertex of $D$ has a positive indegree, then $D$ has a quasi-kernel of size at most $|V|/2$. This conjecture is only confirmed for narrow classes of digraphs, such as semicomplete multipartite, quasi-transitive, or locally semicomplete digraphs. In this note, we state a similar conjecture for all digraphs, show that the two conjectures are equivalent, and prove that both conjectures hold for a class of digraphs containing all orientations of 4-colorable graphs (in particular, of all planar graphs).

Decomposing and colouring some locally semicomplete digraphs

A digraph is semicomplete if any two vertices are connected by at least one arc and is locally semicomplete if the out-neighbourhood and in-neighbourhood of any vertex induce a semicomplete digraph. In this paper we study various subclasses of locally semicomplete digraphs for which we give structural decomposition theorems. As a consequence we obtain several applications, among which an answer to a conjecture of Naserasr and the first and third authors (Aboulker et al., 2021): if an oriented graph is such that the out-neighbourhood of every vertex induces a transitive tournament, then one can partition its vertex set into two acyclic digraphs.

Component Order Connectivity in Directed Graphs

A directed graph D is semicomplete if for every pair x, y of vertices of D, there is at least one arc between x and y. Thus, a tournament is a semicomplete digraph. In the Directed Component Order Connectivity (DCOC) problem, given a digraph D=(V,A) and a pair of natural numbers k and ell , we are to decide whether there is a subset X of V of size k such that the largest strongly connected component in D-X has at most ell vertices. Note that DCOC reduces to the Directed Feedback Vertex Set problem for ell =1. We study the parameterized complexity of DCOC for general and semicomplete digraphs with the following parameters: k, ell ,ell +k and n-ell . In particular, we prove that DCOC with parameter k on semicomplete digraphs can be solved in time O^*(2^{16k}) but not in time O^*(2^{o(k)}) unless the Exponential Time Hypothesis (ETH) fails. The upper bound O^*(2^{16k}) implies the upper bound O^*(2^{16(n-ell )}) for the parameter n-ell . We complement the latter by showing that there is no algorithm of time complexity O^*(2^{o({n-ell })}) unless ETH fails. Finally, we improve (in dependency on ell ) the upper bound of Göke, Marx and Mnich (2019) for the time complexity of DCOC with parameter ell +k on general digraphs from O^*(2^{O(kell log (kell ))}) to O^*(2^{O(klog (kell ))}). Note that Drange, Dregi and van ’t Hof (2016) proved that even for the undirected version of DCOC on split graphs there is no algorithm of running time O^*(2^{o(klog ell )}) unless ETH fails and it is a long-standing problem to decide whether Directed Feedback Vertex Set admits an algorithm of time complexity O^*(2^{o(klog k)}).

Transformation semigroups generated by semicomplete digraphs

Transformation semigroups generated by semicomplete digraphs

On kernels by rainbow paths in arc-coloured digraphs

Abstract In 2018, Bai, Fujita and Zhang [Discrete Math. 341 (2018), no. 6, 1523–1533] introduced the concept of a kernel by rainbow paths (for short, RP-kernel) of an arc-coloured digraph D D , which is a subset S S of vertices of D D such that ( a a ) there exists no rainbow path for any pair of distinct vertices of S S , and ( b b ) every vertex outside S S can reach S S by a rainbow path in D D . They showed that it is NP-hard to recognize whether an arc-coloured digraph has an RP-kernel and it is NP-complete to decide whether an arc-coloured tournament has an RP-kernel. In this paper, we give the sufficient conditions for the existence of an RP-kernel in arc-coloured unicyclic digraphs, semicomplete digraphs, quasi-transitive digraphs and bipartite tournaments, and prove that these arc-coloured digraphs have RP-kernels if certain “short” cycles and certain “small” induced subdigraphs are rainbow.

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.

A Sub-exponential FPT Algorithm and a Polynomial Kernel for Minimum Directed Bisection on Semicomplete Digraphs

Given an n-vertex digraph D and a non-negative integer k, the Minimum Directed Bisection problem asks if the vertices of D can be partitioned into two parts, say L and R, such that \({\vert {L} \vert }\) and \({\vert {R} \vert }\) differ by at most 1 and the number of arcs from R to L is at most k. This problem is known to be NP-hard even when \(k = 0\). We investigate the parameterized complexity of this problem on semicomplete digraphs. We show that Minimum Directed Bisection admits a sub-exponential time fixed-parameter tractable algorithm on semicomplete digraphs. We also show that Minimum Directed Bisection admits a polynomial kernel on semicomplete digraphs. To design the kernel, we use \((n,k,k^2)\)-splitters, which, to the best of our knowledge, have never been used before in the design of kernels. We also prove that Minimum Directed Bisection is NP-hard on semicomplete digraphs, but polynomial time solvable on tournaments.