On the algorithmic complexity of twelve covering and independence parameters of graphs

The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs

Graph classes and forbidden patterns on three vertices

This paper deals with graph classes characterization and recognition. A popular way to characterize a graph class is to list a minimal set of forbidden induced subgraphs. Unfortunately this strategy usually does not lead to an efficient recognition algorithm. On the other hand, many graph classes can be efficiently recognized by techniques based on some interesting orderings of the nodes, such as the ones given by traversals. We study specifically graph classes that have an ordering avoiding some ordered structures. More precisely, we consider what we call patterns on three nodes, and the recognition complexity of the associated classes. In this domain, there are two key previous works. Damashke started the study of the classes defined by forbidden patterns, a set that contains interval, chordal and bipartite graphs among others. On the algorithmic side, Hell, Mohar and Rafiey proved that any class defined by a set of forbidden patterns can be recognized in polynomial time. We improve on these two works, by characterizing systematically all the classes defined sets of forbidden patterns (on three nodes), and proving that among the 23 different classes (up to complementation) that we find, 21 can actually be recognized in linear time. Beyond this result, we consider that this type of characterization is very useful, leads to a rich structure of classes, and generates a lot of open questions worth investigating.Comment: Third version version. 38 page

The Dilworth Number of Auto-Chordal-Bipartite Graphs

The mirror (or bipartite complement) mir(B) of a bipartite graph B=(X,Y,E) has the same color classes X and Y as B, and two vertices x in X and y in Y are adjacent in mir(B) if and only if xy is not in E. A bipartite graph is chordal bipartite if none of its induced subgraphs is a chordless cycle with at least six vertices. In this paper, we deal with chordal bipartite graphs whose mirror is chordal bipartite as well; we call these graphs auto-chordal bipartite graphs (ACB graphs for short). We describe the relationship to some known graph classes such as interval and strongly chordal graphs and we present several characterizations of ACB graphs. We show that ACB graphs have unbounded Dilworth number, and we characterize ACB graphs with Dilworth number k

Regularity of Edge Ideals and Their Powers

We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals of graphs and their powers. Our focus is on bounds and exact values of $\text{ reg } I(G)$ and the asymptotic linear function $\text{ reg } I(G)^q$, for $q \geq 1,$ in terms of combinatorial data of the given graph $G.$Comment: 31 pages, 15 figure

On the Enumeration of Minimal Dominating Sets and Related Notions

A dominating set $D$ in a graph is a subset of its vertex set such that each vertex is either in $D$ or has a neighbour in $D$. In this paper, we are interested in the enumeration of (inclusion-wise) minimal dominating sets in graphs, called the Dom-Enum problem. It is well known that this problem can be polynomially reduced to the Trans-Enum problem in hypergraphs, i.e., the problem of enumerating all minimal transversals in a hypergraph. Firstly we show that the Trans-Enum problem can be polynomially reduced to the Dom-Enum problem. As a consequence there exists an output-polynomial time algorithm for the Trans-Enum problem if and only if there exists one for the Dom-Enum problem. Secondly, we study the Dom-Enum problem in some graph classes. We give an output-polynomial time algorithm for the Dom-Enum problem in split graphs, and introduce the completion of a graph to obtain an output-polynomial time algorithm for the Dom-Enum problem in $P_6$-free chordal graphs, a proper superclass of split graphs. Finally, we investigate the complexity of the enumeration of (inclusion-wise) minimal connected dominating sets and minimal total dominating sets of graphs. We show that there exists an output-polynomial time algorithm for the Dom-Enum problem (or equivalently Trans-Enum problem) if and only if there exists one for the following enumeration problems: minimal total dominating sets, minimal total dominating sets in split graphs, minimal connected dominating sets in split graphs, minimal dominating sets in co-bipartite graphs.Comment: 15 pages, 3 figures, In revisio