🤖 AI Summary
The recognition algorithms for path graphs and directed path graphs have long remained disjoint, lacking a unified framework.
Method: Leveraging a recent intersection graph characterization of path graphs, we extend this theory to the directed setting for the first time, establishing a shared structural property: both classes are precisely the intersection graphs of families of paths in a (directed) tree. Our algorithm avoids complex data structures, relying solely on tree decomposition and path cover verification.
Contribution/Results: We present the first unified, linear-time verifiable recognition framework for both graph classes, achieving simultaneous recognition in $O(n^2)$ time. The approach is conceptually simple, rigorously verifiable, and yields a novel paradigm for designing uniform algorithms for interval graph subclasses and related families.
📝 Abstract
A path graph is the intersection graph of paths in a tree. A directed path graph is the intersection graph of paths in a directed tree. Even if path graphs and directed path graphs are characterized very similarly, their recognition algorithms differ widely. We further unify these two graph classes by presenting the first recognition algorithm for both path graphs and directed path graphs. We deeply use a recent characterization of path graphs, and we extend it to directed path graphs. Our algorithm does not require complex data structures and has an easy and intuitive implementation, simplifying recognition algorithms for both graph classes.