In this paper, we study the enumeration of certain classes of graphs that can be fully characterized by tree decompositions; these classes are particularly significant due to the algorithmic improvements derived from tree decompositions on classically NP-complete problems on these classes [12, 7, 17, 35]. Previously, Chauve et al. [6] and Iriza [26] constructed grammars from the split decomposi...

It is well-known that the GRAPH 3.COLORABILITY problem, deciding whether a given graph has a stable set whose deletion results in a bipartite graph, is NP-complete. We prove the following related theorems: It is NP-complete to decide whether a graph has a stable set whose deletion results in (1) a tree or (2) a trivially perfect graph, and there is a polynomial algorithm to decide if a given gr...

Distance-hereditary graphs form an important class of graphs, from the theoretical point of view, due to the fact that they are the totally decomposable graphs for the split-decomposition. The previous best enumerative result for these graphs is from Nakano et al. (J. Comp. Sci. Tech., 2007), who have proven that the number of distancehereditary graphs on n vertices is bounded by 2d3.59ne. In t...

We consider an unweighted undirected graph with n vertices, m edges, and edge-connectivity 2k. The weak edge orientation problem requires that the edges of this graph be oriented so the resulting directed graph is at least k edge-connected. Nash-Williams proved the existence of such orientations and subsequently Frank [6], Gabow [7], and Nagamochi-Ibaraki [12] gave algorithmic constructions. Al...

We introduce fractional realizations of a graph degree sequence and a closely associated convex polytope. Simple graph realizations correspond to a subset of the vertices of this polytope; we characterize degree sequences for which each polytope vertex corresponds to a simple graph realization. These include the degree sequences of threshold and pseudo-split graphs, and we characterize their re...

Modifying a given graph to obtain another graph is a wellstudied problem with applications in many fields. Given two input graphs G and H, the Contractibility problem is to decide whether H can be obtained from G by a sequence of edge contractions. This problem is known to be NP-complete already when both input graphs are trees of bounded diameter. We prove that Contractibility can be solved in...

A split graph is a whose vertex set can be partitioned into clique and stable set. Given G weight function \(w: V(G) \rightarrow \mathbb {Q}_{\ge 0}\), the Split Vertex Deletion (SVD) problem asks to find minimum of vertices X such that \(G-X\) graph. It easy show if only it does not contain 4-cycle, 5-cycle, or two edge matching as an induced subgraph. Therefore, SVD admits 5-approximation alg...

The families EPT (resp. EPG) Edge Intersection Graphs of Paths in a tree (resp. in a grid) are well studied graph classes. Recently we introduced the graph classes Edge-Intersecting and Non-Splitting Paths in a Tree (ENPT), and in a Grid (ENPG). It was shown that ENPG contains an infinite hierarchy of subclasses that are obtained by restricting the number of bends in the paths. Motivated by thi...

The k-th power H of a graph H is obtained from H by adding new edges between every two distinct vertices having distance at most k in H . Lau [Bipartite roots of graphs, ACM Transactions on Algorithms 2 (2006) 178–208] conjectured that recognizing k-th powers of some graph is NP-complete for all fixed k ≥ 2 and recognizing k-th powers of a bipartite graph is NP-complete for all fixed k ≥ 3. We ...