Trivially perfect graph
In graph theory, a trivially perfect graph is a graph with the property that in each of its induced subgraphs the size of the maximum independent set equals the number of maximal cliques. Trivially perfect graphs were first studied by but were named by ; Golumbic writes that "the name was chosen since it is trivial to show that such a graph is perfect." Trivially perfect graphs are also known as comparability graphs of trees, arborescent comparability graphs, and quasi-threshold graphs. Equivalent characterizations. Trivially perfect graphs have several other equivalent characterizations: Related classes of graphs. It follows  from the equivalent characterizations of trivially perfect graphs that every trivially perfect graph is also a cograph, a chordal graph, a Ptolemaic graph, an interval graph, and a perfect graph. The threshold graphs are exactly the graphs that are both themselves trivially perfect and the complements of trivially perfect graphs (co-trivially perfect graphs). Windmill graphs are trivially perfect. Recognition. describes a simple linear time algorithm for recognizing trivially perfect graphs, based on lexicographic breadth-first search. Whenever the LexBFS algorithm removes a vertex "v" from the first set on its queue, the algorithm checks that all remaining neighbors of  v belong to the same set; if not, one of the forbidden induced subgraphs can be constructed from "v". If this check succeeds for every "v", then the graph is trivially perfect. The algorithm can also be modified to test whether a graph is the complement graph of a trivially perfect graph, in linear time. Determining if a general graph is "k" edge deletions away from a trivially perfect graph is NP-complete, fixed-parameter tractable and can be solved in O(2.45("m" + "n")) time.