Complexity results on the decomposition of a digraph into directed linear forests and out-stars

TL;DR

The paper addresses two digraph-decomposition problems: (k,ell)-BDLFD and (k,ell)-BOGD. It derives complete complexity dichotomies by constructing gadget-based reductions, including reductions to 2-SAT for the simple cases and SAT variants (ME-1-SAT, 3,B2-SAT) for the hard cases, as well as a Hamiltonicity-based reduction for unbounded cases. The results show that (k,ell)-BDLFD is in P when k+ell ≤ 3 and NP-complete otherwise, and (k,ell)-BOGD is in P when min(k,ell) is 1 or infinity, NP-complete in all other cases. The work thus completes the complexity landscape for these two directed-decomposition problems, aligning with undirected analogs and informing algorithmic strategies for directed arboricity-like parameters. It has potential implications for network design and directed-forest decompositions in digraphs.

Abstract

We consider two decomposition problems in directed graphs. We say that a digraph is $k$-bounded for some $k \in \mathbb{Z}_{\geq 1}$ if each of its connected components contains at most $k$ arcs. For the first problem, a directed linear forest is a collection of vertex-disjoint directed paths and we consider the problem of decomposing a given digraph into a $k$-bounded and an $\ell$-bounded directed linear forest for some fixed $k,\ell \in \mathbb{Z}_{\geq 1}\cup \{\infty\}$. We give a full dichotomy for this problem by showing that it can be solved in polynomial time if $k+\ell \leq 3$ and is NP-complete otherwise. This answers a question of Campbell, Hörsch, and Moore. For the second problem, we say that an out-galaxy is a vertex-disjoint collection of out-stars. Again, we give a full dichotomy of when a given digraph can be edge-decomposed into a $k$-bounded and an $\ell$-bounded out-galaxy for fixed $k,\ell \in \mathbb{Z}_{\geq 1}\cup \{\infty\}$. More precisely, we show that the problem can be solved in polynomial time if $\min\{k,\ell\}\in \{1,\infty\}$ and is NP-complete otherwise.

Figures (21)

• Figure 1: A $(k,1)$-decomposition $(F_k,F_1)$ of the short $k$-in-forcer $D$. The dashed red arcs are in $A(F_1)$ and the solid green arcs are in $A(F_k)$.
• Figure 2: A $(k,1)$-decomposition $(F_k,F_1)$ of a long $(k,3)$-in-forcer for some $k \geq 3$ with tip $z$. The triangles indicate short $k$-in-forcers. The solid green arcs are in $A(F_k)$ and the short $k$-in-forcers are decomposed as in Figure \ref{['short1']}.
• Figure 3: An illustration of $D$ and one of its $(k,1)$-decompositions. The squares indicate long $(k,k-2)$-in-forcers. The red dashed arcs are in $A(F_1)$, the solid green arcs are in $A(F_k)$ and the long $(k,k-2)$-in-forcers are decomposed as in Figure \ref{['long']}.
• Figure 4: An illustration of $D$ where the square marks a long $(k,k-3)$-in-forcer. When $k=3$, the square is deleted.
• Figure 5: The different decompositions of $D$. The green solid arcs belong to $A(F_k)$ and the dashed red arcs belong to $A(F_1)$. Each square marks a long $(k,k-3)$-in-forcer, whose decomposition is the one of Figure \ref{['long']}.

Theorems & Definitions (78)

• Theorem 1.1: Nash-Williams NashWilliams1964
• Theorem 1.2: Frank frank1979covering
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Corollary 1.6
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4