Odd-Cycle-Packing Treewidth (OCP-tw)

• OCP-tw is a graph parameter defined by tree decompositions that account for vertex-disjoint odd cycles using apex sets.
• It characterizes graph structure with grid-like parity obstructions and is monotone under odd-minor operations, facilitating efficient solutions for problems like Maximum Independent Set.
• OCP-tw generalizes hybrid width notions and underpins tractable decomposition methods for combinatorial and integer programming problems in odd-minor-closed graph classes.

Odd-Cycle-Packing-treewidth (OCP-tw) is a tree-decomposition-based parameter designed to capture both the structure and algorithmic tractability of graphs in terms of their odd-cycle content. A graph’s OCP-tw quantifies the minimal width needed to decompose it into (possibly overlapping) pieces whose “hardness” derives from the largest number of vertex-disjoint odd cycles present in any single bag. OCP-tw is monotone under odd-minor operations and structurally characterized by grid-like parity obstructions, yielding analogues of key results in minor theory and enabling efficient algorithms for core combinatorial problems such as Maximum Independent Set (MIS) and certain classes of integer programming. This parameter generalizes hybrid width notions, including $\mathcal{B}$-blind-treewidth, and forms the basis of tractable algorithms for odd-minor-closed graph classes (Choi et al., 13 Nov 2025, Gollin et al., 2023).

1. Formal Definition and Basic Properties

Given a finite simple graph $G=(V,E)$, a tree-decomposition is a pair $(T,\beta)$ with $T$ a tree and $\beta:V(T)\to2^V$ satisfying classical covering, edge, and connectedness properties. The OCP-tw refines this notion by introducing apex sets. In an OCP-tree-decomposition $(T,\beta,\alpha)$:

• $\alpha(t)\subseteq\beta(t)$ marks apex vertices in each bag.
• The width of the decomposition is

$$\max\Bigl\{\,\mathrm{adhesion}(T,\beta),\ \max_{t\in V(T)}\bigl(|\alpha(t)|+\mathrm{OCP}(G[\beta(t)\setminus\alpha(t)])\bigr)\Bigr\}.$$

Here, adhesion is the maximum intersection between adjacent bags, and $\mathrm{OCP}(H)$ is the maximal number of vertex-disjoint odd cycles in subgraph $H$. The Odd-Cycle-Packing-treewidth $\mathrm{OCP\text{-}tw}(G)$ is the minimum width over all such decompositions.

OCP-tw is minor-monotone under odd-minors: for any odd-minor $H$ of $G$ (via vertex and edge deletions or bond contractions), $$\mathrm{OCP\text{-}tw}(H)\le\mathrm{OCP\text{-}tw}(G)$$ (Choi et al., 13 Nov 2025, Gollin et al., 2023). Furthermore, $\mathrm{OCP\text{-}tw}(G)\le\mathrm{tw}(G)$ for all graphs.

2. Structural Characterization via Parity Grids and Odd-Minors

OCP-tw is structurally governed by grid-like families of parity obstructions. The two principal infinite grid families are:

• Parity handles $H_k$: a $(4k \times k)$ cylindrical grid with $2k$ antipodal handle-edges, such that every odd cycle must traverse an odd number of these edges.
• Parity vortices $V_k$: similar grids augmented with pairs of “vortex-edges” on inner cycles, forcing odd cycles to intersect vortex pairs oddly.

The OCP Grid Theorem states that there exists a polynomial $f(k)$ such that:

• If $G$ contains $H_k$ or $V_k$ as an odd-minor, then $\mathrm{OCP\text{-}tw}(G)\ge k$.
• If $\mathrm{OCP\text{-}tw}(G)\ge f(k)$, then $G$ contains $H_k$ or $V_k$ as an odd-minor.

Thus, absence of large parity grids certifies bounded OCP-tw. The proof adapts Flat-Wall and society-classification techniques in odd-minor theory to parity-preserving surfaces and tangles (Choi et al., 13 Nov 2025, Gollin et al., 2023).

3. Algorithmic Recognition and Decomposition

A constructive algorithm for OCP-tw proceeds as follows:

1. Search for a large parity-grid odd-minor ($H_k$ or $V_k$) in $G$ using the OCP-Grid search algorithm in time $2^{O(k^c)} n^{O(1)}$.
2. If an obstruction is found, certify $\mathrm{OCP\text{-}tw}(G) > k$.
3. Otherwise, extract an OCP-tree-decomposition of width $f(k) = k^{O(1)}$ (Choi et al., 13 Nov 2025).

This yields a parameterized poly$(k) n^{O(1)}$-time $O(f(k))$-approximation for $\mathrm{OCP\text{-}tw}(G)$. This FPT algorithmic regime stands in contrast to general treewidth for which decomposition is only FPT in width.

4. Connections to Induced Odd-Cycle Packing and Hybrid Widths

OCP-tw subsumes several prior width notions:

• $\mathcal{B}$-blind-treewidth: defined by the minimal width of decompositions over “hard” bags—those containing sets $X$ such that some odd cycle meets $X$ more than once. OCP-tw, via apex sets and packed odd cycles, is precisely equivalent to $\mathcal{B}$-blind-treewidth (Gollin et al., 2023).
• Induced odd-cycle packing numbers: bounds on the number of vertex-disjoint induced odd cycles (iocp$(G)$). For sparse graphs excluding $K_{t,t}$, bounded induced cycle packing number yields logarithmic-treewidth, i.e., $\mathrm{tw}(G)=O_{k,t}(\log n)$ when $iocp(G)<k$ (Bonamy et al., 2022). This supports efficient algorithms in restricted classes.

The connection to odd-minors is fundamental: OCP-tw is monotone under odd-minors and enables decomposition theorems analogous to Robertson–Seymour’s Grid Theorem in the minor universe.

5. Algorithmic Applications: Maximum Independent Set and More

Graphs of bounded OCP-tw admit polynomial-time solutions for the Maximum Independent Set (MIS) problem. Given a width-$k$ OCP-decomposition:

• Perform bottom-up dynamic programming over the tree, tracking at each bag all independent subsets $X\subseteq\beta(t)$.
• Each bag supports $2^{k+1}$ DP states.
• The “apex” guarantees that each torso has OCP$\le k$, permitting polynomial-time solution of local MIS subproblems via known algorithms [Fiorini–Joret–Weltge–Yuditsky].
• Overall runtime is $n^{O(1)}$ for fixed $k$ (Choi et al., 13 Nov 2025).

Analogous decompositional methods extend to Maximum Cut, yielding polynomial or FPT algorithms on odd-minor-closed classes (Gollin et al., 2023). MIS on general graphs parameterized by standard treewidth admits a $2^{O(tw)} n^{O(1)}$ time algorithm, but for OCP-tw, this extends the tractable regime to wider classes.

Additionally, for bounded induced cycle packing in sparse graphs, problems such as MIS, Vertex Cover, Dominating Set, and Coloring become polynomial-time solvable (Bonamy et al., 2022).

6. Extensions to Integer Programming and Signed Graphs

The structural flavor of OCP-tw supports new algorithmic frontiers in integer programming:

• For matrices that are signed incidence matrices of signed graphs $(G,\gamma)$, the OCP-tw parameter extends to “signed OCP-tw,” tracking vertex-disjoint unbalanced cycles (cycles whose edge labels sum to odd).
• Any integer program with constraint matrix of bounded signed OCP-tw—where each bag contains only a bounded number of unbalanced cycles—admits polynomial-time solution for fixed $k$.
• The approach reduces IPs to weighted MIS on signed subdivisions, leverages proximity theorems, and row operations to transform the matrix (Choi et al., 13 Nov 2025).
• Complexity is polynomial in $n$ for fixed $k$ due to bounded proximity and enumeration regimes.

A plausible implication is that OCP-tw underpins tractability for broader classes of $\Delta$-modular matrices and other labeled graph frameworks.

7. Parameter Bounds, Lower Bounds, and Open Problems

The explicit bounds on OCP-tw arise from composed structural theorems in odd-minor theory, with all functions $f(k)$ elementary-recursive (compositions of Grohe’s, Thomassen’s, and barycentric bounds). There is, to date, no improved lower bound against such blow-up except general minor-width hardness (Gollin et al., 2023).

For graph packing parameterized by treewidth, general odd-cycle packing (finding a maximal set of vertex-disjoint $C_\ell$ for fixed odd $\ell$) admits an algorithm running in $2^{O(tw \log tw)} n^{O(1)}$ time, and there is a matching lower bound under ETH: no algorithm can run in $2^{o(tw \log tw)} n^{O(1)}$ time (Esmer et al., 7 Sep 2025).

Open directions include complexity dichotomies for MIS and Max-Cut on broader odd-minor-free classes, characterizing half-integral Erdős–Pósa relations for odd-minors, and improving parameter compositions for practical decomposition routines (Gollin et al., 2023). The extension of OCP-tw to directed, group-labeled, or matrix-labeled frameworks remains an active field of paper.

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Key References:

• "Odd-Cycle-Packing-treewidth: On the Maximum Independent Set problem in odd-minor-free graph classes" (Choi et al., 13 Nov 2025)
• "Structure and algorithms for graphs excluding grids with small parity breaks as odd-minors" (Gollin et al., 2023)
• "Sparse graphs with bounded induced cycle packing number have logarithmic treewidth" (Bonamy et al., 2022)
• "Generalized Graph Packing Problems Parameterized by Treewidth" (Esmer et al., 7 Sep 2025)