Subclasses of k-trees: Characterization and recognition

A clique-difference encoding scheme for labelled k-path graphs

“…K -path graphs were first introduced in [7], along with a characterization theorem concerning the existence and exact number of simplicial vertices. Based on Theorem 4, the recognition of a k-tree G = (V , E) as a k-path graph is straightforward: either it is a complete graph with k + 1 vertices or it has exactly two vertices with degree k, which are simplicial in G. In both cases, it suffices to examine |Adj(v)|, ∀v ∈ V .…”

“…In [7], k-path graphs were defined and characterized: the family extends path graphs in the same way that k-trees do it for ordinary trees. In this paper we show that every k-path graph with n vertices can be assigned a sequence of n − k − 1 pairs of vertices, named the reduced sequence and we investigate its structural properties.…”

The mean order of sub‐k‐trees of k‐trees

“…This is the critical number of edges beyond that a graph of order n fails to be planar. For a characterization of the class of planar 3‐trees see . For all k‐trees are nonplanar, with the exception of the complete graph K 4 that happens to have edges.…”

“…It is this property that allows us to reduce such a k ‐tree in a useful way. The subclass in question has been previously studied (see ) under the name simple‐clique k ‐trees , and so we adopt the same terminology and provide the same recursive definition. Definition (Simple‐clique (SC) k ‐trees) Fix an integer .…”

“…Thus if is a perfect elimination ordering of a path‐type k ‐tree, then the subgraph induced by is a k ‐clique. The class of path‐type k ‐trees has previously been studied (see ) under the name k‐path graphs . Definition (Star‐type k ‐trees) Fix an integer .…”