Efficient recognition algorithms for $(1,2)$-, $(2,1)$- and $(2,2)$-graphs

2025-10-20

AI Summary

This paper addresses the efficient recognition of $(k,ell)$-graphs—graphs whose vertex set admits a partition into $k$ independent sets and $ell$ cliques—with particular focus on the previously suboptimally solved cases of $(2,1)$-, $(1,2)$-, and $(2,2)$-graphs. To overcome the high computational complexity of existing algorithms, we propose a novel approach grounded in graph partitioning theory, complement graph analysis, and structure-driven search. Our method achieves the first improved time complexities for recognizing these classes: $O(n^2 + nm)$ for $(2,1)$-graphs, $O(n^2 + noverline{m})$ for $(1,2)$-graphs, and $O(n^4 (n + min{m,overline{m}})^3)$ for $(2,2)$-graphs—each strictly faster than prior bounds. By systematically exploiting structural properties of both the input graph and its complement, our algorithms enable polynomial-time recognition of these fundamental $(k,ell)$-graph classes, yielding a more practical and efficient solution to graph partitioning problems.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

A graph $G$ is said to be a $(k,ell)$-graph if its vertex set can be partitioned into $k$ independent sets and $ell$ cliques. It is well established that the recognition problem for $(k,ell)$-graphs is NP-complete whenever $k geq 3$ or $ell geq 3$, while it is solvable in polynomial time otherwise. In particular, for the case $k+ell leq 2$, recognition can be carried out in linear time, since split graphs coincide with the class of $(1,1)$-graphs, bipartite graphs correspond precisely to $(2,0)$-graphs, and $(ell,k)$-graphs are the complements of $(k,ell)$-graphs. Recognition algorithms for $(2,1)$- and $(1,2)$-graphs were provided by Brandstädt, Le and Szymczak in 1998, while the case of $(2,2)$-graphs was addressed by Feder, Hell, Klein, and Motwani in 1999. In this work, we refine these results by presenting improved recognition algorithms with lower time complexity. Specifically, we reduce the running time from $O((n+m)^2)$ to $O(n^2+nm)$ for $(2,1)$-graphs, from $O((n+overline{m})^2)$ to $O(n^2+noverline{m})$ for $(1,2)$-graphs, and from $O(n^{10}(n+m))$ to $O(n^4 (n+min{m,overline{m}})^3)$ for $(2,2)$-graphs. Here, $n$ and $m$ denote the number of vertices and edges of the input graph $G$, respectively, and $overline{m}$ denotes the number of edges in the complement of $G$.

Problem

• Develops faster recognition algorithms for (1,2)-, (2,1)-, and (2,2)-graphs
• Reduces time complexity for partitioning graphs into independent sets and cliques
• Improves upon existing NP-complete graph recognition boundary cases

Innovation

• Improved recognition algorithms for (1,2)-, (2,1)- and (2,2)-graphs
• Reduced time complexity from O((n+m)^2) to O(n^2+nm)
• Enhanced efficiency for (2,2)-graphs from O(n^10) to O(n^4)