Hedge Intersection Graph: Structures & Complexity

• Hedge Intersection Graph is a structure that encodes overlap patterns among hedges in a partitioned graph, influencing algorithmic tractability.
• Its acyclicity enables polynomial-time solutions for Hedge Cluster Deletion, while the presence of cycles introduces NP-hard challenges.
• Distinct from traditional line and hypergraph intersection graphs, it maps edge-partition overlaps to vertices, guiding combinatorial optimization.

A hedge intersection graph is an abstract structure that encodes the intersection patterns among specified edge subsets—known as “hedges”—in an underlying graph. Formally, for a graph $H$ whose edge set is partitioned into hedges $\mathcal{E} = \{E_1,\ldots,E_\ell\}$, the hedge intersection graph $\mathcal{F}$ has one vertex per hedge and edges corresponding to nontrivial overlaps of the hedges in $H$ at vertices. This concept appears prominently in the paper of structural and algorithmic properties of combinatorial optimization problems on graphs with partitioned edge sets, such as the Hedge Cluster Deletion problem. The structure and complexity of the hedge intersection graph directly impact tractability, as acyclicity (forest structure) permits polynomial-time exact algorithms for certain deletion and clustering tasks, while cycles introduce NP-hardness. The notion is distinct from intersection graphs of connected subgraphs (such as $H$-topological intersection graphs) but is closely related to hypergraph intersection and line graph constructs.

1. Definition and Construction

Let $H = (V, \mathcal{E})$ be a hedge graph: a simple undirected graph with its edge set $E(H)$ partitioned into $\ell$ groups called hedges, $\mathcal{E} = \{E_1, \ldots, E_\ell\}$, where each $E_i \subseteq E(H)$. The hedge intersection graph $\mathcal{F} = (\mathcal{E}, E(\mathcal{F}))$ is defined by:

• $V(\mathcal{F}) = \mathcal{E}$, i.e., each vertex corresponds to a hedge.
• Two distinct hedges $E_i$ and $E_j$ are adjacent in $\mathcal{F}$ if and only if there exists a vertex $v \in V$ and (not necessarily distinct) edges $e \in E_i$, $f \in E_j$ such that $e$ and $f$ both contain $v$.

Equivalently, $E(\mathcal{F}) = \big\{\,\{i,j\} : i \neq j$ and $\exists\,uv\in E_i$, $vw\in E_j$ for some $v\in V\,\big\}$.

Construction of $\mathcal{F}$ from $H$ proceeds by:

• Enumerating all unordered hedge pairs $\{i, j\}$.
• For each such pair, scanning edges in $E_i \cup E_j$ to determine whether the overlap condition is satisfied.
• Complexity is $O(\ell \cdot |E(H)|)$ naively, or $O(n^3+\ell^2)$ with optimizations, where $|E(H)|=O(n^2)$ for $n=|V|$ (Konstantinidis et al., 13 Nov 2025).

2. Structural Properties: Acyclicity and Triangles

A hedge intersection graph is acyclic if it contains no cycles, i.e., it is a forest. The absence of a $3$-cycle in $\mathcal{F}$ precisely reflects the fact that no triangle in $H$ is covered by three distinct hedges; every triangle in $H$ must be covered by at most two hedges in this case. Each component of an acyclic $\mathcal{F}$ is a tree, and edges of $\mathcal{F}$ can be classified as witnessing either a path $P_3$ (inducing a chain of three vertices) or a triangle $K_3$ in $H$, or both. These classifications are critical in algorithmic reductions.

3. Algorithmic Implications: Cluster Deletion

When $\mathcal{F}$ is acyclic, the Hedge Cluster Deletion problem—removing a minimum number of hedges so the remaining graph is a union of cliques—admits a polynomial-time solution. This is achieved via reduction to a minimum vertex cover in a bipartite graph constructed from $\mathcal{F}$ and the underlying hinge structure. The process exploits domination structures and partitionings in $\mathcal{F}$, using key lemmas:

• Lemma 5.2: The domination sets $R(x)$ for each hedge $x$ can be computed in polynomial time.
• Lemma 5.3: All hedges spanning an internal $P_3$ or certain mixed vertices must be in any solution.
• Lemma 5.4: Each tree component in $\mathcal{F}$ (after appropriate deletions) is connected by at most one "cross-edge" to another component.
• Lemma 5.5/5.6: The constructed bipartite graph $F_2$ encodes the necessary constraints, and its minimum vertex cover exactly matches feasible hedge-deletion choices.
• Theorem 5.8: This process yields a polynomial-time algorithm whenever $\mathcal{F}$ is acyclic.

In contrast, as soon as $\mathcal{F}$ contains a cycle, the interaction of hedges can encode NP-hard problems (e.g., 3-vertex path hitting corresponding to hypergraph edge cover), eliminating prospects of efficient exact solutions in the general case (Konstantinidis et al., 13 Nov 2025).

4. Relation to Other Intersection Graph Notions

The hedge intersection graph is specialized to edge-partitioned graphs, distinct from the $H$-topological intersection graphs ($H$-graphs) or general hypergraph intersection graphs (line graphs). For instance, the line graph or hyperedge intersection graph (as in the oriented hypergraph setting (Reff, 2015)) considers vertices or hyperedges as nodes, with intersections corresponding to shared endpoints or vertices. In contrast, the hedge intersection graph's vertices are the hedges (partition classes), with edges determined by overlap at vertices in the underlying graph. This places the hedge intersection graph in a distinct intersection-theoretic framework, with unique combinatorial constraints arising from hedge structure.

5. Complexity Classifications and Applications

The tractability of hedge-based combinatorial problems is closely governed by the topology of the hedge intersection graph:

• If $\mathcal{F}$ is acyclic, Hedge Cluster Deletion can be solved in polynomial time (Konstantinidis et al., 13 Nov 2025).
• If $\mathcal{F}$ contains cycles, the problem becomes NP-hard or hard to approximate within certain exponential factors.
• In practical terms, for classes of graphs in which every triangle is covered by at most two hedges, structural properties of $\mathcal{F}$ allow constant-factor approximation algorithms even outside the fully acyclic case.
• Other related optimization problems, such as constrained variants of vertex cover (subject to hedge-specific constraints), are incorporated as subroutines in these algorithms.

6. Illustrative Example and Limiting Case

Suppose hedges $E_1$, $E_2$, $E_3$ in $H$ form a cycle in $\mathcal{F}$, each pair $E_i$, $E_{i+1}$ (indices modulo $3$) overlapping at a vertex to form a three-edge cycle. Any minimal hedge-deletion set to obtain a graph as a disjoint union of cliques must hit all such overlaps; the problem reduces to hitting all edges in a triangle of a hypergraph, which is NP-hard. In contrast, if $\mathcal{F}$ is a tree, overlapping components can be managed independently after cut-edge removals (Konstantinidis et al., 13 Nov 2025).

7. Connections and Distinctions with Edge Intersection Hypergraphs

There is a potential for confusion between hedge intersection graphs and other intersection-derived graphs on edge groups. In the edge intersection hypergraph $EI(\mathcal{H})$ of a hypergraph $\mathcal{H}$, the edge set is composed of all pairwise intersections of hyperedges of size at least two (Sonntag et al., 2019). This is structurally distinct from the hedge intersection graph, which records overlaps at vertices between classes (hedges) rather than forming new edge sets from intersections. The two notions serve different roles: one for representing adjacency of parts (hedges), the other for generating hyperedges from overlaps.

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The hedge intersection graph provides a detailed and algorithmically critical representation of how edge groupings (hedges) in a partitioned graph interact. Its structural properties, especially acyclicity, directly determine the computational complexity of associated optimization problems, and its distinctions from topological and hypergraph intersection frameworks clarify its unique position in the combinatorial landscape.