Strongly connected orientations and integer lattices

Abstract

Let $D=(V,A)$ be a digraph whose underlying undirected graph is $2$-edge-connected, and let $P$ be the polytope whose vertices are the incidence vectors of arc sets whose reversal makes $D$ strongly connected. We study the lattice theoretic properties of the integer points contained in a proper face $F$ of $P$ not contained in $\{x:x_a=i\}$ for any $a\in A,i\in \{0,1\}$. We prove under a mild necessary condition that $F\cap \{0,1\}^A$ contains an integral basis $B$, i.e., $B$ is linearly independent, and any integral vector in the linear hull of $F$ is an integral linear combination of $B$. This result is surprising as the integer points in $F$ do not necessarily form a Hilbert basis. In proving the result, we develop a theory similar to Matching Theory for degree-constrained dijoins in bipartite digraphs. Our result has consequences for head-disjoint strong orientations in hypergraphs, and also to a famous conjecture by Woodall that the minimum size of a dicut of $D$, say $τ$, is equal to the maximum number of disjoint dijoins. We prove a relaxation of this conjecture, by finding for any prime number $p\geq 2$, a $p$-adic packing of dijoins of value $τ$ and of support size at most $2|A|$. We also prove that the all-ones vector belongs to the lattice generated by $F\cap \{0,1\}^A$, where $F$ is the face of $P$ satisfying $x(δ^+(U))=1$ for every dicut $δ^+(U)$ with minimum size.

Figures (7)

• Figure 1: A digraph $D=(V,A)$ (left). The set $\delta^+(\{1,2\})=\{(1,4),(2,3)\}$ is a dicut of $D$. The subset $\{(1,4)\}$ is a strengthening set of $D$, as reversing the arc in the set yields a strongly connected digraph (middle). The subset $\{(1,4),(2,3)\}$ is a dijoin of $D$, as bidirecting the arcs in the set yields a strongly connected digraph (right).
• Figure 2: (Left) A digraft $(D,\mathcal{F})$ where $\mathcal{F}$ consists of $V\setminus v$ for all sinks $v$, as well as $U=\{2,3,7,8\}$. The set of bold arcs is a dijoin $J$ whose indicator vector belongs to $F(D,\mathcal{F})$. (Middle/Right) A digraft $(D,\mathcal{F})$ where $\mathcal{F}$ consists of $V\setminus v$ for all sinks $v$, and $\{u\}$ for every filled-in source $u$. The middle digraft is balanced basic, and robust. The right digraft is skewed basic, and robust. In all three digrafts, the filled-in nodes are tight, and the non-filled-in nodes are active.
• Figure 3: (Left) A digraph $D$. The set $J$ of bold arcs forms a strengthening set whose indicator vector belongs to $F$, for $\mathcal{F}=\emptyset$. (Right) The digraft $(D',\mathcal{F}')$ constructed in the proof of \ref{['scr->dij']}. The filled-in nodes are new, while the non-filled-in ones are the original nodes. The set $\phi(J)$ of bold arcs is the image of $J$ under the mapping: it is a dijoin whose indicator vector belongs to $F(D',\mathcal{F}')$.
• Figure 4: (Left) A basic non-robust digraft $(D,\mathcal{F})$ where $\mathcal{F}$ consists only of $V\setminus v$ for all sinks $v$. (Right) A affine critical digraft $(D,\mathcal{F})$ where $\mathcal{F}$ consists only of $V\setminus v$ for all sinks $v$. This digraft is not basic. In both figures, the filled-in/non-filled-in correspond to the tight/active nodes.
• Figure 5: (Left) A digraft $(D,\mathcal{F})$ where $\mathcal{F}$ consists of $V\setminus v$ for all sinks $v$. Denote by $J$ the set of bold arcs. (Right: top, bottom) The two $(U,V\setminus U)$-contractions $(D_i,\mathcal{F}_i),i=1,2$. Denote by $J_i$ the set of bold arcs of $(D_i,\mathcal{F}_i)$ for $i=1,2$.

Theorems & Definitions (74)

• Theorem 1.1
• Conjecture 1.2: Schrijver-note
• Theorem 1.3
• Theorem 1.4
• Conjecture 1.5
• Theorem 1.6
• Definition 1.7: bipartite digraph
• Definition 1.8: digraft
• Theorem 1.9
• Definition 1.10: balanced/skewed basic digraft