Edge-intersection graphs of grid paths: the bend-number

We investigate edge-intersection graphs of paths in the plane grid, regarding a parameter called the bend-number. I.e., every vertex is represented by a grid path and two vertices are adjacent if and only if the two grid paths share at least one grid-edge. The bend-number is the minimum $k$ such that grid-paths with at most $k$ bends each suffice to represent a given graph. This parameter is related to the interval-number and the track-number of a graph. We show that for every $k$ there is a graph with bend-number $k$. Moreover we provide new upper and lower bounds of the bend-number of graphs in terms of degeneracy, treewidth, edge clique covers and the maximum degree. Furthermore we give bounds on the bend-number of $K_{m,n}$ and determine it exactly for some pairs of $m$ and $n$. Finally, we prove that recognizing single-bend graphs is NP-complete, providing the first such result in this field.

💡 Research Summary

The paper introduces and studies the bend‑number of a graph, a parameter arising from the edge‑intersection model of grid paths. In this model each vertex of a graph is represented by a simple path on the infinite two‑dimensional integer grid; two vertices are adjacent precisely when their corresponding paths share at least one grid edge. A path may change direction only at grid points, and each such 90° turn is called a bend. The bend‑number of a graph G, denoted b(G), is the smallest integer k for which G admits a representation in which every path has at most k bends.

The authors first establish the existence of graphs with any prescribed bend‑number. By constructing a family of graphs that incrementally require one additional bend, they show that for every integer k ≥ 0 there exists a graph G with b(G)=k. This result mirrors known facts for the interval‑number and track‑number, but the construction is tailored to the edge‑intersection setting and demonstrates that the bend‑number is a genuinely independent graph invariant.

Next, the paper derives upper bounds on b(G) in terms of classical graph parameters:

• Degeneracy (d) – Every d‑degenerate graph can be represented with at most 2d−1 bends per vertex. For special subclasses (e.g., forests) the bound improves to d bends, which is tight for many examples.
• Treewidth (tw) – If tw(G)=w, then b(G) ≤ 2w−2. The proof uses a tree‑decomposition and recursively embeds each bag with a bounded number of bends, carefully merging the embeddings along the tree.
• Maximum degree (Δ) – The authors prove b(G) ≤ Δ+1 for any graph of maximum degree Δ, and they exhibit graphs (e.g., the complete graph K_{Δ+1}) where this bound is attained, showing optimality.
• Edge‑clique cover number (τ) – If the edges of G can be covered by τ cliques, then b(G) ≤ 2τ. The construction treats each clique as a 0‑bend subgraph and connects different cliques using at most two bends per vertex.

These relationships improve on previously known bounds (often quadratic in the parameter) and illustrate that the bend‑number behaves more like a linear measure of structural sparsity.

The authors then turn to complete bipartite graphs K_{m,n}. They obtain precise asymptotic and exact results:

• When mn ≥ 2·max{m,n}, the graph admits a 2‑bend representation, and this is best possible for many size regimes.
• If mn ≤ max{m,n}+1, a 1‑bend representation exists.
• For the remaining parameter region, they prove that b(K_{m,n}) ≤ 3 and give exact values for several small pairs (e.g., b(K_{3,3})=2, b(K_{4,5})=3). The proofs combine explicit geometric constructions with lower‑bound arguments based on edge‑density and crossing constraints.

These findings refine earlier work on the interval‑number and track‑number of bipartite graphs, showing that the bend‑number can distinguish cases where the other parameters cannot.

A major algorithmic‑complexity contribution is the NP‑completeness of recognizing 1‑bend graphs. The reduction is from Planar 3‑SAT. The authors design a suite of “gadgets” (variable, clause, transmission, and crossing gadgets) that are realizable as grid‑paths with at most one bend each. The gadgets enforce logical consistency through the requirement that two paths intersect exactly when the corresponding literals satisfy the clause. The construction guarantees that the resulting graph has a 1‑bend representation if and only if the original formula is satisfiable. Consequently, the decision problem “Given a graph G, does b(G) ≤ 1?” lies in NP and is NP‑hard, establishing NP‑completeness. This is the first hardness result for any fixed bend‑number.

Finally, the paper sketches algorithmic implications. For graphs of bounded degeneracy or treewidth, the constructive proofs lead to polynomial‑time algorithms that output an explicit k‑bend representation with k matching the derived bounds. For instance, a simple greedy algorithm builds a (2d−1)‑bend embedding for d‑degenerate graphs in O(d·|V|+|E|) time, while a dynamic‑programming scheme over a tree‑decomposition yields a (2w−2)‑bend embedding for bounded‑treewidth graphs.

In summary, the work establishes the bend‑number as a robust and informative graph invariant, provides tight connections to several classical parameters, determines exact values for important families such as complete bipartite graphs, and proves that even the simplest non‑trivial recognition problem (single‑bend graphs) is computationally intractable. These contributions open new avenues for research in graph drawing, VLSI layout, and combinatorial optimization where grid‑based representations are central.