The Even-Path Problem in Directed Single-Crossing-Minor-Free Graphs

TL;DR

This work studies the directed EvenPath problem, seeking a simple $s$-$t$ path of even length in directed graphs. It proves polynomial-time solvability for $H$-minor-free graphs when $H$ is a fixed single-crossing graph, by combining a Robertson–Seymour $3$-clique-sum decomposition with parity-preserving reductions and a parity-aware disjoint-path framework. The authors introduce parity-mimicking networks that preserve parity configurations for up to three terminals, enabling branch compression while maintaining planarity or bounded treewidth; they also develop a polynomial-time algorithm for a four-terminal disjoint-path with total parity problem in planar graphs (under specific constraints) that supports phase-two projections. The approach yields a two-phase algorithm: Phase I collapses branch pieces into parity-mimicking networks to produce a tractable core, and Phase II propagates parity information through a chain of pieces via projection networks to decide the existence of an $s$-$t$ even path, with a self-reduction providing an actual path when a solution exists. Together, these results extend tractability from planar graphs to a broad minor-closed family and suggest new techniques for parity-preserving reductions in directed graphs.

Abstract

Finding a simple path of even length between two designated vertices in a directed graph is a fundamental NP-complete problem known as the EvenPath problem. Nedev proved in 1999, that for directed planar graphs, the problem can be solved in polynomial time. More than two decades since then, we make the first progress in extending the tractable classes of graphs for this problem. We give a polynomial time algorithm to solve the EvenPath problem for classes of H-minor-free directed graphs,1 where H is a single-crossing graph. We make two new technical contributions along the way, that might be of independent interest. The first, and perhaps our main, contribution is the construction of small, planar, parity-mimicking networks. These are graphs that mimic parities of all possible paths between a designated set of terminals of the original graph. Finding vertex disjoint paths between given source-destination pairs of vertices is another fundamental problem, known to be NP-complete in directed graphs, though known to be tractable in planar directed graphs. We encounter a natural variant of this problem, that of finding disjoint paths between given pairs of vertices, but with constraints on parity of the total length of paths. The other significant contribution of our paper is to give a polynomial time algorithm for the 3-disjoint paths with total parity problem, in directed planar graphs with some restrictions (and also in directed graphs of bounded treewidth).

Figures (7)

• Figure 1: An example of a graph $G$. We ignore directions here.
• Figure 2: A clique sum decomposition of $G$. Red nodes are the clique nodes and blue node the piece nodes. Dashed edges denote virtual edges.
• Figure 3: Figure a) shows the input graph and b) shows the graph with $L$ replaced by an erroneous mimicking network $L'$. Suppose the original graph in a) has no $s$-$t$ even path but does have an $s$-$t$ even walk as shown in the figure, using vertex $v_2$ twice. If we query for a path from $v_1$ to $v_3$ in $L$, and add a direct $v_1$ to $v_3$ path of that parity in $L'$, we end up creating a false solution since $v_2$ is freed up to be used outside $L'$. Hence there must be equality between corresponding direct sets.
• Figure 4: Figure a) denotes the original graph which has both a direct path, as well as a via path of even parity from $v_1$ to $v_3$. Suppose the via path is part of an even $s$-$t$ path solution, as marked by blue. Then in $L$ itself, we could replace the via path by the direct path and it would still be a valid even $s$-$t$ path, as marked in blue in b). Hence in the mimicking network $L'$, too (shown in c)), we could use the direct $v_1$ to $v_3$ path of the same parity. Therefore we do not need to put the parity of the $v_1$-$v_2$-$v_3$ path in $\text{Via}_{L}(v_1,v_2,v_3)$, since the same parity is already present in $\text{Dir}_{L}(v_1,v_3)$, and $\text{Dir}_{L}(v_1,v_3) = \text{Dir}_{L'}(v_1,v_3)$.
• Figure 5: Fig a) denotes a graph $L$ with its parity configuration table (only relevant sets). Fig b) denotes a 'parity mimicking network', if for each pair of terminals, we just independently put paths of correct parity, disjoint from each other. It leads to an extra path (highlighted in red) from $v_1$ to $v_3$ via $v_2$ in $L'$, of odd parity. Pairs of such entries, for which we cannot add disjoint paths are called bad entries as marked by the dashed red line in the parity configuration table in a). Fig c) outlines the approach used to construct the correct mimicking network. The two paths corresponding to bad pair entries, form the bad kernel, for which we construct a mimicking network by enumerating cases. The remaining paths can be added iteratively, disjoint from all existing paths, on the outer face.

Theorems & Definitions (14)

• Theorem 1
• Definition 2
• Theorem 3: Robertson-Seymour RS93
• Definition 4
• Definition 5
• Lemma 5
• Lemma 5
• Definition 6
• Lemma 6
• Definition 7