Abstract. We perform a systematic study in the computational complexity of the connected variant of three related transversal problems: Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. Just like their original counterparts, these variants are NP-complete for general graphs. A graph G is H-free for some graph H if G contains no induced subgraph isomorphic to H. It is known that Connected Vertex Cover is NP-complete even for H-free graphs if H contains a claw or a cycle. We show that the two other connected variants also remain NP-complete if H contains a cycle or claw. In the remaining case H is a linear forest. We show that Connected Vertex Cover, Connected Feedback Vertex Set, and Connected Odd Cycle Transversal are polynomial-time solvable for sP2-free graphs for every constant s ≥ 1. For proving these results we use known results on the price of connectivity for vertex cover, feedback vertex set, and odd cycle transversal. This is the first application of the price of connectivity that results in polynomial-time algorithms.
We study the class of 1-perfectly orientable graphs, that is, graphs having an orientation in which every out-neighborhood induces a tournament. 1-perfectly orientable graphs form a common generalization of chordal graphs and circular arc graphs. Even though they can be recognized in polynomial time, little is known about their structure. In this paper, we develop several results on 1-perfectly orientable graphs. In particular, we: (i) give a characterization of 1-perfectly orientable graphs in terms of edge clique covers, (ii) identify several graph transformations preserving the class of 1-perfectly orientable graphs, (iii) exhibit an infinite family of minimal forbidden induced minors for the class of 1-perfectly orientable graphs, and (iv) characterize the class of 1-perfectly orientable graphs within the classes of cographs and of cobipartite graphs. The class of 1-perfectly orientable co-bipartite graphs coincides with the class of co-bipartite circular arc graphs.
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Ho proved in [A note on the total domination number, Util. Math. 77 (2008) 97-100] that the total domination number of the Cartesian product of any two graphs without isolated vertices is at least one half of the product of their total domination numbers. We extend a result of Lu and Hou from [Total domination in the Cartesian product of a graph and K 2 or C n , Util. Math. 83 (2010) 313-322] by characterizing the pairs of graphs G and H for which γ t (G H) = 1 2 γ t (G)γ t (H) , whenever γ t (H) = 2. In addition, we present an infinite family of graphs G n with γ t (G n ) = 2n, which asymptotically approximate the equality in γ t (G n G n ) ≥ 1 2 γ t (G n ) 2 .
A graph G is said to be 1-perfectly orientable if it has an orientation such that for every vertex v ∈ V (G), the out-neighborhood of v in D is a clique in G. In 1982, Skrien posed the problem of characterizing the class of 1-perfectly orientable graphs. This graph class forms a common generalization of the classes of chordal and circular arc graphs; however, while polynomially recognizable via a reduction to 2-SAT, no structural characterization of this intriguing class of graphs is known. Based on a reduction of the study of 1-perfectly orientable graphs to the biconnected case, we characterize, both in terms of forbidden induced minors and in terms of composition theorems, the classes of 1-perfectly orientable K 4 -minor-free graphs and of 1-perfectly orientable outerplanar graphs. As part of our approach, we introduce a class of graphs defined similarly as the class of 2-trees and relate the classes of graphs under consideration to two other graph classes closed under induced minors studied in the literature: cyclically orientable graphs and graphs of separability at most 2. *
A graph G is said to be 1-perfectly orientable (1-p.o. for short) if it admits an orientation such that the out-neighborhood of every vertex is a clique in G. The class of 1-p.o. graphs forms a common generalization of the classes of chordal and circular arc graphs. Even though 1-p.o. graphs can be recognized in polynomial time, no structural characterization of 1-p.o. graphs is known. In this paper we consider the four standard graph products: the Cartesian product, the strong product, the direct product, and the lexicographic product. For each of them, we characterize when a nontrivial product of two graphs is 1-p.o.The concept of 1-p.o. graphs was introduced in 1982 by Skrien [16] (under the name {B 2 }graphs), where the problem of characterizing 1-p.o. graphs was posed. While a structural understanding of 1-p.o. graphs is still an open question, partial results are known. Bang-Jensen et al. observed in [1] that 1-p.o. graphs can be recognized in polynomial time via a reduction to 2-SAT. Skrien [16] characterized graphs admitting an orientation that is both an in-tournament and an out-tournament as exactly the proper circular arc graphs. All chordal graphs and all circular arc graphs are 1-p.o. [17], and, more generally, so is any vertex-intersection graph of connected induced subgraphs of a unicyclic graph [1,15]. Every graph having a unique induced cycle of order at least 4 is 1-p.o. [1].In [8], several operations preserving the class of 1-p.o. graphs were described (see Sec. 2); operations that do not preserve the property in general were also considered. In the same paper 1-p.o. graphs were characterized in terms of edge-clique covers, and characterizations of 1p.o. cographs and of 1-p.o. co-bipartite graphs were given. In particular, a cograph is 1-p.o. if and only if it is K 2,3 -free and a co-bipartite graph is 1-p.o. if and only if it is circular arc. A structural characterization of line graphs that are 1-p.o. was given in [1].In this paper we consider the four standard graph products: the Cartesian product, the strong product, the direct product, and the lexicographic product. For each of these four products, we completely characterize when a nontrivial product of two graphs G and H is 1-p.o. While the results for the Cartesian, the lexicographic, and the direct products turn out to be rather straightforward, the characterization for the case of the strong product is more involved.Some common features of the structure of the factors involved in the characterizations can be described as follows. In the cases of the Cartesian and the direct product the factors turn out to be very sparse and very restricted, always having components with at most one cycle. In the case of the lexicographic and of the strong product the factors can be dense. More specifically, co-bipartite 1-p.o. graphs, including co-chain graphs in the case of strong products, play an important role in these characterizations. The case of the strong product also leads to a new infinite family of 1-p.o. graphs (cf. Proposition 6.8).The pa...
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