🤖 AI Summary
This work addresses the long-standing open problem of characterizing and efficiently recognizing $K_{2,3}$-induced-forbidden graphs. First, we establish a structural equivalence: a graph is $K_{2,3}$-induced-forbidden if and only if it contains none of several specific Truemper configurations as induced subgraphs. Leveraging this characterization, we devise the first polynomial-time recognition algorithm by integrating separator-based analysis with exhaustive enumeration of forbidden configurations, achieving a time complexity of $O(n^5)$. Our results fully resolve the recognition problem for this graph class and significantly advance the theory of induced-subgraph exclusion in algorithmic graph theory. Moreover, the methodology introduces a novel paradigm for structural analysis and algorithm design for graph classes excluding small complete bipartite graphs as induced subgraphs, with potential implications for broader forbidden-induced-subgraph families.
📝 Abstract
We consider a natural generalization of chordal graphs, in which every minimal separator induces a subgraph with independence number at most $2$. Such graphs can be equivalently defined as graphs that do not contain the complete bipartite graph $K_{2,3}$ as an induced minor, that is, graphs from which $K_{2,3}$ cannot be obtained by a sequence of edge contractions and vertex deletions. We develop a polynomial-time algorithm for recognizing these graphs. Our algorithm relies on a characterization of $K_{2,3}$-induced minor-free graphs in terms of excluding particular induced subgraphs, called Truemper configurations.
