The connectivity carcass of a vertex subset in a graph: both odd and even case

TL;DR

The paper delivers the first complete, self-contained exposition of the connectivity carcass for storing all ${S}$-mincuts in undirected multigraphs, uniting odd- and even-capacity cases under a unified submodularity-based framework. It introduces three core components—the flesh ${\mathcal F}$, the skeleton ${\mathcal H}$, and the projection mapping ${\pi}$—to represent all ${S}$-mincuts in ${O}(m)$ space, while enabling efficient query answering via ${O}(m)$-time strips and ${O}(n)$-space projections. A simpler approach based on cut submodularity replaces prior reliance on locally orientable graphs, and a detailed mapping between ${\mathcal F}$ and ${\mathcal H}$ yields efficient, scalable data structures for reporting and separating $S$-mincuts. These results provide a practical foundation for related problems, including dynamic all-pairs mincuts, by exploiting the structured interplay between flesh, skeleton, and projection. Overall, the work clarifies the connectivity carcass machinery and strengthens its potential applications in network design and analysis.

Abstract

Let $G=(V,E)$ be an undirected unweighted multi-graph and $S\subseteq V$ be a subset of vertices. A set of edges with the least cardinality whose removal disconnects $S$, that is, there is no path between at least one pair of vertices from $S$, is called a Steiner mincut for $S$ or simply an $S$-mincut. Connectivity Carcass is a compact data structure storing all $S$-mincuts in $G$ announced by Dinitz and Vainshtein in an extended abstract by Dinitz and Vainshtein in 1994. The complete proof of various results of this data structure for the simpler case when the capacity of $S$-mincut is odd appeared in the year 2000 in SICOMP. Over the last couple of decades, there have been attempts towards the proof for the case when the capacity of $S$-mincut is even, but none of them met a logical end. We present the following results. - We present the first complete, self-contained exposition of the connectivity carcass which covers both even and odd cases of the capacity of $S$-mincut. - We derive the results using an alternate and much simpler approach. In particular, we derive the results using submodularity of cuts -- a well-known property of graphs expressed using a simple inequality. - We also show how the connectivity carcass can be helpful in efficiently answering some basic queries related to $S$-mincuts using some additional insights.

Figures (28)

• Figure 1: Each vertex, other than $s$ and $t$, has the same number of incoming and outgoing edges.
• Figure 2: ($i$) vertices $\nu,\mu$, and $w$ have degrees $\ge 4$, ($ii$) inflating $\nu$ and $\mu$, $(iii)$ implanting cycles at $\nu$ and $\mu$.
• Figure 3: There may be many $S$-mincuts associated with one minimal cuts in ${\cal H}$.
• Figure 4: ($i$) the partitions of $S$ that distinguish $\omega_1$ shown in distinct colors ($ii$) the corresponding cuts in ${\cal H}$.
• Figure 5: ($i$) ${\cal F}$, ($ii$) Projection mapping of all non-$S$-units in skeleton ${\cal H}$.

Theorems & Definitions (131)

• Definition 1: Single-source single-sink balanced DAG
• Definition 2: Quotient graph induced by a family of cuts
• Theorem 1: Theorem 4.4 in DBLP:conf/soda/DinitzV95
• Theorem 2
• Definition 3: Cut
• Definition 4: Intersection and union of cuts
• Definition 5: Dominance
• Definition 6: Minimum cut
• Definition 7: Minimal cut
• Definition 8: ($s,t$)-cut