holes in graph theory). Every chordal graph is a strangulated graph, a graph in which every peripheral cycle is a triangle, because peripheral cycles are a special case of induced cycles. Strangulated graphs are graphs that can be formed by clique-sums of chordal graphs and maximal planar graphs. Therefore, strangulated graphs include maximal planar graphs. Chordal completions and treewidth If is an arbitrary graph, a chordal completion of (or minimum fill-in) is a chordal graph that contains as a subgraph. The parameterized version of minimum fill-in is fixed parameter tractable, and moreover, is solvable in parameterized subexponential time. The treewidth of is one less than the number of vertices in a maximum clique of a chordal completion chosen to minimize this clique size. The -trees are the graphs to which no additional edges can be added without increasing their treewidth to a number larger than . Therefore, the -trees are their own chordal completions, and form a subclass of the chordal graphs. Chordal completions can also be used to characterize several other related classes of graphs. Notes References . . . . . . . . . . . . . . . . . . . External links Information System on Graph Class Inclusions: chordal graph Graph families Perfect graphs Intersection classes of graphs