Khái niệm cốt lõi
The author presents a unified recognition algorithm for both path graphs and directed path graphs, which simplifies and unifies the study of these two graph classes from an algorithmic perspective.
Tóm tắt
The content discusses the recognition of path graphs and directed path graphs, which are classes of graphs between interval graphs and chordal graphs. The key points are:
Path graphs are the intersection graphs of paths in a tree, while directed path graphs are the intersection graphs of paths in a directed tree. Even though these two graph classes are characterized very similarly, their recognition algorithms differ widely.
The author presents the first recognition algorithm that can handle both path graphs and directed path graphs. The algorithm is based on a recent characterization of path graphs by Apollonio and Balzotti, which is extended to directed path graphs.
The proposed algorithm has a simpler and more intuitive implementation compared to previous algorithms for path graphs and directed path graphs. It does not require complex data structures and unifies the recognition of these two graph classes.
For path graphs, the algorithm provides a simpler treatment compared to previous algorithms, while for directed path graphs, it is the only one that does not rely on the results from Chaplick et al. that establish a linear-time algorithm to determine if a path graph is also a directed path graph.
The time complexity of the proposed algorithm is O(p(m+n)), where p is the number of cliques in the input graph, and m and n are the number of edges and vertices, respectively.