Satgraphs and independent domination. Part 1

AbstractA graph G is called a satgraph if there exists a partition A∪B=V(G) such that•A induces a clique [possibly, A=∅],•B induces a matching [i.e., G(B) is a 1-regular subgraph, possibly, B=∅], and•there are no triangles (a,b,b′), where a∈A and b,b′∈B.We also introduce the hereditary closure of SAT, denoted by HSAT [hereditary satgraphs]. The class HSAT contains split graphs. In turn, HSAT is contained in the class of all (1,2)-split graphs [A. Gyárfás, Generalized split graphs and Ramsey numbers, J. Combin. Theory Ser. A 81 (2) (1998) 255–261], the latter being still not characterized. We characterize satgraphs in terms of forbidden induced subgraphs.There exist close connections between satgraphs and the satisfiability problem [SAT]. In fact, SAT is linear-time equivalent to finding the independent domination number in the corresponding satgraph. It follows that the independent domination problem is NP-complete for the hereditary satgraphs. In particular, it is NP-complete for perfect graphs

Independent domination in hereditary classes

AbstractWe investigate Independent Domination Problem within hereditary classes of graphs. Boliac and Lozin [Independent domination in finitely defined classes of graphs, Theoret. Comput. Sci. 301 (1–3) (2003) 271–284] proved some sufficient conditions for Independent Domination Problem to be NP-complete within finitely defined hereditary classes of graphs. They posed a question whether the conditions are also necessary. We show that the conditions are not necessary, since Independent Domination Problem is NP-hard within 2P3-free graphs.Moreover, we show that the problem remains NP-hard for a new hereditary class of graphs, called hereditary 3-satgraphs. We characterize hereditary 3-satgraphs in terms of forbidden induced subgraph. As corollaries, we prove that Independent Domination Problem is NP-hard within the class of all 2P3-free perfect graphs and for K1,5-free weakly chordal graphs.Finally, we compare complexity of Independent Domination Problem with that of Independent Set Problem for a hierarchy of hereditary classes recently proposed by Hammer and Zverovich [Construction of maximal stable sets with k-extensions, Combin. Probab. Comput. 13 (2004) 1–8]. For each class in the hierarchy, a maximum independent set can be found in polynomial time, and the hierarchy covers all graphs. However, our characterization of hereditary 3-satgraphs implies that Independent Domination Problem is NP-hard for almost all classes in the hierarchy. This fact supports a conjecture that Independent Domination is harder than Independent Set Problem within hereditary classes

Independent domination versus weighted independent domination.

Independent domination is one of the rare problems for which the complexity of weighted and unweighted versions is known to be different in some classes of graphs. Trying to better understand the gap between the two versions of the problem, in the present paper we prove two complexity results. First, we extend NP-hardness of the weighted version in a certain class to the unweighted case. Second, we strengthen polynomial-time solvability of the unweighted version in the class of -free graphs to the weighted case. This result is tight in the sense that both versions are NP-hard in the class of -free graphs, i.e. is a minimal graph forbidding of which produces an NP-hard case for both versions of the problem

More results on weighted independent domination

Weighted independent domination is an NP-hard graph problem, which remains computationally intractable in many restricted graph classes. In particular, the problem is NP-hard in the classes of sat-graphs and chordal graphs. We strengthen these results by showing that the problem is NP-hard in a proper subclass of the intersection of sat-graphs and chordal graphs. On the other hand, we identify two new classes of graphs where the problem admits polynomial-time solutions

Cliques, colouring and satisfiability : from structure to algorithms

We examine the implications of various structural restrictions on the computational complexity of three central problems of theoretical computer science (colourability, independent set and satisfiability), and their relatives. All problems we study are generally NP-hard and they remain NP-hard under various restrictions. Finding the greatest possible restrictions under which a problem is computationally difficult is important for a number of reasons. Firstly, this can make it easier to establish the NP-hardness of new problems by allowing easier transformations. Secondly, this can help clarify the boundary between tractable and intractable instances of the problem. Typically an NP-hard graph problem admits an infinite sequence of narrowing families of graphs for which the problem remains NP-hard. We obtain a number of such results; each of these implies necessary conditions for polynomial-time solvability of the respective problem in restricted graph classes. We also identify a number of classes for which these conditions are sufficient and describe explicit algorithms that solve the problem in polynomial time in those classes. For the satisfiability problem we use the language of graph theory to discover the very first boundary property, i.e. a property that separates tractable and intractable instances of the problem. Whether this property is unique remains a big open problem

Treewidth versus clique number. II. Tree-independence number

In 2020, we initiated a systematic study of graph classes in which the treewidth can only be large due to the presence of a large clique, which we call $(\mathrm{tw},\omega)$-bounded. While $(\mathrm{tw},\omega)$-bounded graph classes are known to enjoy some good algorithmic properties related to clique and coloring problems, it is an interesting open problem whether $(\mathrm{tw},\omega)$-boundedness also has useful algorithmic implications for problems related to independent sets. We provide a partial answer to this question by means of a new min-max graph invariant related to tree decompositions. We define the independence number of a tree decomposition $\mathcal{T}$ of a graph as the maximum independence number over all subgraphs of $G$ induced by some bag of $\mathcal{T}$. The tree-independence number of a graph $G$ is then defined as the minimum independence number over all tree decompositions of $G$. Generalizing a result on chordal graphs due to Cameron and Hell from 2006, we show that if a graph is given together with a tree decomposition with bounded independence number, then the Maximum Weight Independent Packing problem can be solved in polynomial time. Applications of our general algorithmic result to specific graph classes will be given in the third paper of the series [Dallard, Milani\v{c}, and \v{S}torgel, Treewidth versus clique number. III. Tree-independence number of graphs with a forbidden structure].Comment: 33 pages; abstract has been shortened due to arXiv requirements. A previous version of this arXiv post has been reorganized into two parts; this is the first of the two parts (the second one is arXiv:2206.15092