Hereditary biclique-Helly graphs: recognition and maximal biclique enumeration

A biclique is a set of vertices that induce a bipartite complete graph. A graph G is biclique-Helly when its family of maximal bicliques satisfies the Helly property. If every induced subgraph of G is also biclique-Helly, then G is hereditary biclique-Helly. A graph is C_4-dominated when every cycle of length 4 contains a vertex that is dominated by the vertex of the cycle that is not adjacent to it. In this paper we show that the class of hereditary biclique-Helly graphs is formed precisely by those C_4-dominated graphs that contain no triangles and no induced cycles of length either 5, or 6. Using this characterization, we develop an algorithm for recognizing hereditary biclique-Helly graphs in O(n^2+\alpha m) time and O(m) space. (Here n, m, and \alpha = O(m^{1/2}) are the number of vertices and edges, and the arboricity of the graph, respectively.) As a subprocedure, we show how to recognize those C_4-dominated graphs that contain no triangles in O(\alpha m) time and O(m) space. Finally, we show how to enumerate all the maximal bicliques of a C_4-dominated graph with no triangles in O(n^2 + \alpha m) time and O(\alpha m) space, and we discuss how some biclique problems can be solved in O(\alpha m) time and O(n+m) space.

💡 Research Summary

The paper studies graphs whose maximal bicliques satisfy the Helly property and, moreover, whose every induced subgraph also enjoys this property – the hereditary biclique‑Helly graphs. Building on the earlier characterization by Groshaus and Szwarcfiter, which listed six forbidden induced subgraphs (triangles, C5, C6, and the “ladder” graphs), the authors introduce a more algorithm‑friendly description: a graph is hereditary biclique‑Helly if and only if it is C4‑dominated and free of triangles, C5 and C6. A C4‑dominated graph is one in which every induced 4‑cycle contains a pair of vertices that are comparable by domination (i.e., one’s neighbourhood is contained in the other’s). This new structural insight enables a substantial improvement in both recognition and enumeration algorithms.

Recognition algorithm.
The algorithm proceeds in four steps.

1. Detect a triangle using the Chiba–Nishizeki method, which runs in O(m α) time (α is the arboricity, bounded by √m).
2. Test whether the graph is C4‑dominated. The authors encode all 4‑cycles compactly in a “square family” data structure: for each ordered pair (v,w) with v > w they store the set L(v,w) of common neighbours that are smaller than v. Each triple (v,w,L) represents all 4‑cycles where v is the highest‑ordered vertex. A cycle is safe if v dominates w; otherwise it is unsafe and must be checked that the vertices of L(v,w) are totally ordered by domination. By scanning the square family and a pre‑computed domination matrix D, a non‑dominated C4 can be found (or the graph is declared C4‑dominated) in O(α m) time and O(m) space.
3. Search for an induced C5, and 4. search for an induced C6. Both are performed by bounded‑depth DFS/BFS, each taking O(n²) time in the worst case.

If none of the four forbidden configurations appear, the graph is hereditary biclique‑Helly. The total running time is O(n² + α m) and the space consumption is O(m), a dramatic reduction from the previously known O(n³ m²) bound.

Enumeration of maximal bicliques.
For a C4‑dominated triangle‑free graph, the authors prove that every maximal biclique can be expressed as (A(v), B(v)) for some vertex v, where A(v) consists of vertices adjacent to v or that dominate v, and B(v) consists of vertices adjacent to v or that are dominated by v. Consequently, the number of maximal bicliques never exceeds n. The enumeration algorithm simply constructs A(v) and B(v) for each vertex, filters duplicates by processing vertices in a degree‑ordering, and outputs the resulting bicliques. Building the adjacency and domination lists costs O(m + α m) = O(α m) time; the overall enumeration runs in O(n² + α m) time (the n² term accounts for output size) and uses O(α m) additional memory.

Further algorithmic consequences.
Because domination relations can be computed in O(α m) time, several related problems become linear‑time in the arboricity: finding a minimum dominating set, a maximum independent biclique, and a minimum biclique cover can all be solved within the same asymptotic bounds (O(α m) time, O(n + m) space).

Technical contributions.

1. A clean structural equivalence: hereditary biclique‑Helly ⇔ C4‑dominated ∧ {triangle, C5, C6}-free.
2. The square‑family representation, which stores all 4‑cycles in O(m α) space despite the potential O(n²) number of cycles.
3. An O(n² + α m) recognition algorithm that improves upon the earlier O(n³ m²) approach.
4. A proof that maximal bicliques in the target class are bounded by n and a linear‑time enumeration method.
5. Extensions to other biclique‑related optimization problems with the same time/space guarantees.

Impact and applications.
The results are particularly relevant for large‑scale network analysis where bicliques model dense bipartite interactions (e.g., user–item recommendation, gene–disease associations). The ability to test hereditary biclique‑Helly property efficiently enables preprocessing steps for algorithms that rely on Helly‑type guarantees. Moreover, the linear‑time enumeration of maximal bicliques offers a practical tool for data mining tasks that require exhaustive biclique extraction without exponential blow‑up. The arboricity‑based complexity makes the algorithms well‑suited for sparse graphs, which are common in real‑world datasets. Overall, the paper delivers both a deeper theoretical understanding of hereditary biclique‑Helly graphs and concrete, scalable algorithms that bridge the gap between theory and practice.