Composition and product of cover-incomparability graphs

Listing, generating or enumerating objects of specified type is one of the principal tasks in algorithmics. In graph algorithms one often enumerates vertex subsets satisfying a certain property. We study the enumeration of all minimal connected dominating sets of an input graph from various graph classes of bounded chordality. We establish enumeration algorithms as well as lower and upper bounds for the maximum number of minimal connected dominating sets in such graphs. In particular, we present algorithms to enumerate all minimal connected dominating sets of chordal graphs in time O(1.7159^n), of split graphs in time O(1.3803^n), and of AT-free, strongly chordal, and distance-hereditary graphs in time O^*(3^{n/3}), where n is the number of vertices of the input graph. Our algorithms imply corresponding upper bounds for the number of minimal connected dominating sets for these graph classes.

Motivated by the problem of reconstructing evolutionary history, Nishimura, Radge and Thilikos introduced the notion of k-leaf powers as the class of graphs G = (V, E) which have a k-leaf root, i.e., a tree T with leaf set V where xy ∈ E if and only if the T-distance between x and y is at most k. It is known that leaf powers are strongly chordal (i.e., sun-free chordal) graphs. Despite extensive research, the problem of recognizing leaf powers, i.e., to decide for a given graph G whether it is a k-leaf power for some k, remains open. Much less is known on the complexity of finding the leaf rank of G, i.e., to determine the minimum number k such that G is a k-leaf power. A result by Bibelnieks and Dearing implies that not every strongly chordal graph is a leaf power. Recently, Kennedy, Lin and Yan have shown that dart- and gem-free chordal graphs are 4-leaf powers. We generalize their result and show that ptolemaic (i.e., gem-free chordal) graphs are leaf powers. Moreover, ptolemaic graphs have unbounded leaf rank. Furthermore, we show that interval graphs are leaf powers which implies that leaf powers have unbounded clique-width. Finally, we characterize unit interval graphs as those leaf powers having a caterpillar leaf root.

A graph with a vertex set is said to be a prime graph if there exists a bijective mapping , where denotes the number of vertices in , such that for any two adjacent vertices and in have . Tensor Product graph is a way to combine (compose) two graphs into one larger and more complex graph. The result is a new graph that reflects the connection properties of the two original graphs, but in a very specific and more complex way than other graph operations. Therefore, this research aims to determine whether there is prime labeling in the class of graphs resulting from the Tensor Product of the path graph and the cycle graph . The research employed analytical and exploratory methods with a trial-and-error strategy to determine the labeling that possesses a prime property. The results of this study prove that two classes of the Tensor Product graph for , and graph , for are prime graph. This finding expands the results on classes of graphs that admit prime labeling and provides a basis for further research on graph labeling in other graph operations

Laminar structure of ptolemaic graphs with applications

Fiber-complemented graphs — I: structure and invariant subgraphs

Consider a class of graphs $$\mathcal{G}$$ having a polynomial time algorithm computing the set of all minimal separators for every graph in $$\mathcal{G}$$ . We show that there is a polynomial time algorithm for treewidth and minimum fill-in, respectively, when restricted to the class $$\mathcal{G}$$ . Many interesting classes of intersection graphs have a polynomial time algorithm computing all minimal separators, like permutation graphs, circle graphs, circular arc graphs, distance hereditary graphs, chordal bipartite graphs etc. Our result generalizes earlier results for the treewidth and minimum fill-in for several of these classes. We also consider the related problems pathwidth and interval completion when restricted to some special graph classes.

Linear-time algorithms for three domination-based separation problems in block graphs

For a family of graphs F, given a graph G and an integer k, the F-Deletion problem asks whether we can delete at most k vertices from G to obtain a graph in the family F. The F-Deletion problems for all non-trivial families F that satisfy the hereditary property on induced subgraphs are known to be NP-hard by a result of Yannakakis (STOC'78). Ptolemaic graphs are the graphs that satisfy the Ptolemy inequality, and they are the intersection of chordal graphs and distance-hereditary graphs. Equivalently, they form the set of graphs that do not contain any chordless cycles or a gem as an induced subgraph. (A gem is the graph on 5 vertices, where four vertices form an induced path, and the fifth vertex is adjacent to all the vertices of this induced path.) The Ptolemaic Deletion problem is the F-Deletion problem, where F is the family of Ptolemaic graphs. In this paper we study Ptolemaic Deletion from the viewpoint of Kernelization Complexity, and obtain a kernel with 𝒪(k⁶) vertices for the problem.

We introduce a new subclass of chordal graphs that generalizes the class of split graphs, which we call well-partitioned chordal graphs. A connected graph G is a well-partitioned chordal graph if there exist a partition P of the vertex set of G into cliques and a tree T having P as a vertex set such that for distinct X,Y∈P, (1) the edges between X and Y in G form a complete bipartite subgraph whose parts are some subsets of X and Y, if X and Y are adjacent in T, and (2) there are no edges between X and Y in G otherwise. A split graph with vertex partition (C,I) where C is a clique and I is an independent set is a well-partitioned chordal graph as witnessed by a star T having C as the center and each vertex in I as a leaf, viewed as a clique of size 1. We characterize well-partitioned chordal graphs by forbidden induced subgraphs, and give a polynomial-time algorithm that given a graph, either finds an obstruction, or outputs a partition of its vertex set that asserts that the graph is well-partitioned chordal.We observe that there are problems, for instance Densestk-Subgraph and b-Coloring, that are polynomial-time solvable on split graphs but become NP -hard on well-partitioned chordal graphs. On the other hand, we show that the Geodetic Set problem, known to be NP -hard on chordal graphs, can be solved in polynomial time on well-partitioned chordal graphs.We also answer two combinatorial questions on well-partitioned chordal graphs that are open on chordal graphs, namely that each well-partitioned chordal graph admits a polynomial-time constructible tree 3-spanner, and that each (2-connected) well-partitioned chordal graph has a vertex that intersects all its longest paths (cycles).

Bipartite bithreshold graphs

We provide polynomial-time algorithms for counting the number of perfect matchings in chain graphs, cochain graphs, and threshold graphs. These algorithms are based on newly developed subdivision schemes that we call a recursive decomposition. On the other hand, we show the $\sharp$ P-completeness for counting the number of perfect matchings in chordal graphs, split graphs and chordal bipartite graphs. This is in an interesting contrast with the fact that counting the number of independent sets in chordal graphs can be done in linear time.

On dynamic threshold graphs and related classes

On polygon numbers of circle graphs and distance hereditary graphs

The edge-orientation problem and some of its variants on weighted graphs

In the past decades for more and more graph classes the Graph Isomorphism Problem was shown to be solvable in polynomial time. An interesting family of graph classes arises from intersection graphs of geometric objects. In this work we show that the Graph Isomorphism Problem for unit square graphs, intersection graphs of axis-parallel unit squares in the plane, can be solved in polynomial time. Since the recognition problem for this class of graphs is NP-hard we can not rely on standard techniques for geometric graphs based on constructing a canonical realization. Instead, we develop new techniques which combine structural insights into the class of unit square graphs with understanding of the automorphism group of such graphs. For the latter we introduce a generalization of bounded degree graphs which is used to capture the main structure of unit square graphs. Using group theoretic algorithms we obtain sufficient information to solve the isomorphism problem for unit square graphs.