Strong arc decompositions of split digraphs

A {\bf strong arc decomposition} of a digraph $D=(V,A)$ is a partition of its arc set $A$ into two sets $A_1,A_2$ such that the digraph $D_i=(V,A_i)$ is strong for $i=1,2$. Bang-Jensen and Yeo (2004) conjectured that there is some $K$ such that every $K$-arc-strong digraph has a strong arc decomposition. They also proved that with one exception on 4 vertices every 2-arc-strong semicomplete digraph has a strong arc decomposition. Bang-Jensen and Huang (2010) 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 {\bf split digraph} is a digraph whose underlying undirected graph is a split graph, meaning that its vertices can be partioned into a clique and an independent set. Equivalently, a split digraph is any digraph which can be obtained from a semicomplete digraph $D=(V,A)$ by adding a new set $V'$ of vertices and some arcs between $V'$ and $V$. 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

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

Spanning eulerian subdigraphs in semicomplete digraphs

International audienceA digraph is eulerian if it is connected and every vertex has its in-degree equal to its outdegree. Having a spanning eulerian subdigraph is thus a weakening of having a hamiltonian cycle. In this paper, we first characterize the pairs (D, a) of a semicomplete digraph D and an arc a such that D has a spanning eulerian subdigraph containing a. In particular, we show that if D is 2-arc-strong, then every arc is contained in a spanning eulerian subdigraph. We then characterize the pairs (D, a) of a semicomplete digraph D and an arc a such that D has a spanning eulerian subdigraph avoiding a. 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 f (k) such that every f (k)-arc-strong semicomplete digraph contains a spanning eulerian subdigraph avoiding any prescribed set of k arcs. We conjecture that f (k) = k + 1 and establish this conjecture for k ≤ 3 and when the k arcs that we delete form a forest of stars. A digraph D is eulerian-connected if for any two distinct vertices x, y, the digraph D has a spanning (x, y)-trail. We prove that every 2-arc-strong semicomplete digraph is eulerianconnected. All our results may be seen as arc analogues of well-known results on hamiltonian paths and cycles in semicomplete digraphs

Spanning eulerian subdigraphs in semicomplete digraphs

A digraph is eulerian if it is connected and every vertex has its in-degree equal to its outdegree. Having a spanning eulerian subdigraph is thus a weakening of having a hamiltonian cycle. In this paper, we first characterize the pairs (D, a) of a semicomplete digraph D and an arc a such that D has a spanning eulerian subdigraph containing a. In particular, we show that if D is 2-arc-strong, then every arc is contained in a spanning eulerian subdigraph. We then characterize the pairs (D, a) of a semicomplete digraph D and an arc a such that D has a spanning eulerian subdigraph avoiding a. 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 f (k) such that every f (k)-arc-strong semicomplete digraph contains a spanning eulerian subdigraph avoiding any prescribed set of k arcs. We conjecture that f (k) = k + 1 and establish this conjecture for k ≤ 3 and when the k arcs that we delete form a forest of stars. A digraph D is eulerian-connected if for any two distinct vertices x, y, the digraph D has a spanning (x, y)-trail. We prove that every 2-arc-strong semicomplete digraph is eulerianconnected. All our results may be seen as arc analogues of well-known results on hamiltonian paths and cycles in semicomplete digraphs

Detours in Directed Graphs

We study two "above guarantee" versions of the classical Longest Path problem on undirected and directed graphs and obtain the following results. In the first variant of Longest Path that we study, called Longest Detour, the task is to decide whether a graph has an (s,t)-path of length at least dist_G(s,t)+k (where dist_G(s,t) denotes the length of a shortest path from s to t). Bezáková et al. [Ivona Bezáková et al., 2019] proved that on undirected graphs the problem is fixed-parameter tractable (FPT) by providing an algorithm of running time 2^{O(k)}⋅ n. Further, they left the parameterized complexity of the problem on directed graphs open. Our first main result establishes a connection between Longest Detour on directed graphs and 3-Disjoint Paths on directed graphs. Using these new insights, we design a 2^{O (k)}· n^{O(1)} time algorithm for the problem on directed planar graphs. Further, the new approach yields a significantly faster FPT algorithm on undirected graphs. In the second variant of Longest Path, namely Longest Path above Diameter, the task is to decide whether the graph has a path of length at least diam(G)+k(diam(G)denotes the length of a longest shortest path in a graph G). We obtain dichotomy results about Longest Path above Diameter on undirected and directed graphs. For (un)directed graphs, Longest Path above Diameter is NP-complete even for k=1. However, if the input undirected graph is 2-connected, then the problem is FPT. On the other hand, for 2-connected directed graphs, we show that Longest Path above Diameter is solvable in polynomial time for each k ∈ {1,..., 4} and is NP-complete for every k ≥ 5. The parameterized complexity of Longest Detour on general directed graphs remains an interesting open problem.publishedVersio