Polynomial Time Recognition of Squares of Ptolemaic Graphs and 3-sun-free Split Graphs
The square of a graph \(G\), denoted \(G^2\), is obtained from \(G\) by putting an edge between two distinct vertices whenever their distance is two. Then \(G\) is called a square root of \(G^2\). Deciding whether a given graph has a square root is known to be NP-complete, even if the root is required to be a chordal graph or even a split graph.
We present a polynomial time algorithm that decides whether a given graph has a ptolemaic square root. If such a root exists, our algorithm computes one with a minimum number of edges.
In the second part of our paper, we give a characterization of the graphs that admit a 3-sun-free split square root. This characterization yields a polynomial time algorithm to decide whether a given graph has such a root, and if so, to compute one.