Odd-Cycle-Packing-treewidth: On the Maximum Independent Set problem in odd-minor-free graph classes

TL;DR

This paper introduces Odd-Cycle-Packing-treewidth (OCP-tw), a width parameter that blends tree-like structure with the distribution of odd cycles in a graph, and proves an odd-minor analogue of the Grid Theorem using parity grids. The authors show MIS is solvable in time n^{f(k)} on graphs with OCP-tw ≤ k and present a constructive framework to either certify bounded OCP-tw or produce obstructions, extending to integer programs with totally Δ-modular incidence-like matrices. The approach unifies MIS tractability across odd-minor-free classes and provides a modular toolkit (surfaces, renditions, society classifications, and parity-aware walls) that yields explicit polynomial bounds and algorithmic procedures. The results have implications for parameterized MIS and related IP contexts, offering a robust path to tractable instances beyond traditional treewidth by exploiting parity and minor-structure together. Overall, the work advances structural graph theory and algorithmic applications by delivering constructive decompositions, parity-aware obstructions, and a bridge to IPs via signed graph and Δ-modular frameworks.

Abstract

We introduce the tree-decomposition-based graph parameter Odd-Cycle-Packing-treewidth (OCP-tw) as a width parameter that asks to decompose a given graph into pieces of bounded odd cycle packing number. The parameter OCP-tw is monotone under the odd-minor-relation and we provide an analogue to the celebrated Grid Theorem of Robertson and Seymour for OCP-tw. That is, we identify two infinite families of grid-like graphs whose presence as odd-minors implies large OCP-tw and prove that their absence implies bounded OCP-tw. This structural result is constructive and implies a 2^(poly(k))poly(n)-time parameterized poly(k)-approximation algorithm for OCP-tw. Moreover, we show that the (weighted) Maximum Independent Set problem (MIS) can be solved in polynomial time on graphs of bounded OCP-tw. Finally, we lift the concept of OCP-tw to a parameter for matrices of integer programs. To this end, we show that our strategy can be applied to efficiently solve integer programs whose matrices can be "tree-decomposed" into totally delta-modular matrices with at most two non-zero entries per row.

Figures (9)

• Figure 1: Representatives of our two obstructing families for OCP-treewidth: The parity handle of order $5$ (left) and the parity vortex of order $5$ (right). Notice that a cycle in either of these graphs is odd if and only if it contains an odd number of the red edges.
• Figure 2: Representatives of three more types of parity grids: The single parity break of order $5$ (left), a grid with odd cycle outgrows of order $5$ (middle), and a parity crosscap, also known as an Escher grid, of order $5$ (right). As before, a cycle in either of these graphs is odd if and only if it contains an odd number of the red edges.
• Figure 3: The complexity landscape of MIS in odd-minor-closed graph classes. The bottom shows the two instances of $K_t$-odd-minor-free classes where MIS is in P. The two middle areas depict odd-minor-monotone parameters and the corresponding (parameterized) complexity classes for MIS and finally, the very top indicates the general regime of $K_t$-odd-minor-free graphs for $t\geq 5$ where MIS is known to be NP-hard and to admit a PTAS. An arrow from $x$ to $y$ indicates that membership in the class $x$, or having the parameter $x$ bounded also implies membership in the class $y$ or that the parameter $y$ is bounded.
• Figure 4: We remove edges of $\mathscr{H}_{k^2}$ to obtain $W'$ whose maximum degree is $3$. The paths $Q_\ell'$ are coloured red.
• Figure 5: A bipartite $(6\times 6)$-mesh.

Theorems & Definitions (106)

• Definition 1.1: OCP-treewidth
• Theorem 1.2: see thm:miswithboundedocptw
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.2
• proof
• Lemma 2.3
• proof