k-Efficient domination: Algorithmic perspective

A set [Formula: see text] of a graph [Formula: see text] is called an efficient dominating set of [Formula: see text] if every vertex [Formula: see text] has exactly one neighbor in [Formula: see text], in other words, the vertex set [Formula: see text] is partitioned to some circles with radius one such that the vertices in [Formula: see text] are the centers of partitions. A generalization of this concept, introduced by Chellali et al. [k-Efficient partitions of graphs, Commun. Comb. Optim. 4 (2019) 109–122], is called [Formula: see text]-efficient dominating set that briefly partitions the vertices of graph with different radiuses. It leads to a partition set [Formula: see text] such that each [Formula: see text] consists a center vertex [Formula: see text] and all the vertices in distance [Formula: see text], where [Formula: see text]. In other words, there exist the dominators with various dominating powers. The problem of finding minimum set [Formula: see text] is called the minimum [Formula: see text]-efficient domination problem. Given a positive integer [Formula: see text] and a graph [Formula: see text], the [Formula: see text]-efficient Domination Decision problem is to decide whether [Formula: see text] has a [Formula: see text]-efficient dominating set of cardinality at most [Formula: see text]. The [Formula: see text]-efficient Domination Decision problem is known to be NP-complete even for bipartite graphs [M. Chellali, T. W. Haynes and S. Hedetniemi, k-Efficient partitions of graphs, Commun. Comb. Optim. 4 (2019) 109–122]. Clearly, every graph has a [Formula: see text]-efficient dominating set but it is not correct for efficient dominating set. In this paper, we study the following: [Formula: see text]-efficient domination problem set is NP-complete even in chordal graphs. A polynomial-time algorithm for [Formula: see text]-efficient domination in trees. [Formula: see text]-efficient domination on sparse graphs from the parametrized complexity perspective. In particular, we show that it is [Formula: see text]-hard on d-degenerate graphs while the original dominating set has Fixed Parameter Tractable (FPT) algorithm on d-degenerate graphs. [Formula: see text]-efficient domination on nowhere-dense graphs is FPT.

On the Parameterized Approximability of Contraction to Classes of Chordal Graphs

A graph operation that contracts edges is one of the fundamental operations in the theory of graph minors. Parameterized Complexity of editing to a family of graphs by contracting k edges has recently gained substantial scientific attention, and several new results have been obtained. Some important families of graphs, namely, the subfamilies of chordal graphs, in the context of edge contractions, have proven to be significantly difficult than one might expect. In this article, we study the F -Contraction problem, where F is a subfamily of chordal graphs, in the realm of parameterized approximation. Formally, given a graph G and an integer k , F -Contraction asks whether there exists X ⊆ E(G) such that G/X ∈ F and | X | ≤ k . Here, G/X is the graph obtained from G by contracting edges in X . We obtain the following results for the F - Contraction problem: • Clique Contraction is known to be FPT . However, unless NP⊆ coNP/ poly , it does not admit a polynomial kernel. We show that it admits a polynomial-size approximate kernelization scheme ( PSAKS ). That is, it admits a (1 + ε)-approximate kernel with O ( k f(ε)) vertices for every ε > 0. • Split Contraction is known to be W[1]-Hard . We deconstruct this intractability result in two ways. First, we give a (2+ε)-approximate polynomial kernel for Split Contraction (which also implies a factor (2+ε)- FPT -approximation algorithm for Split Contraction ). Furthermore, we show that, assuming Gap-ETH , there is no (5/4-δ)- FPT -approximation algorithm for Split Contraction . Here, ε, δ > 0 are fixed constants. • Chordal Contraction is known to be W[2]-Hard . We complement this result by observing that the existing W[2]-hardness reduction can be adapted to show that, assuming FPT ≠ W[1] , there is no F(k) - FPT -approximation algorithm for Chordal Contraction . Here, F(k) is an arbitrary function depending on k alone. We say that an algorithm is an h(k) - FPT -approximation algorithm for the F -Contraction problem, if it runs in FPT time, and on any input (G, k) such that there exists X ⊆ E(G) satisfying G/X ∈ F and | X | ≤ k , it outputs an edge set Y of size at most h(k) ċ k for which G/Y is in F .

Chordality in Matroids: In Search of the Converse to Hliněný's Theorem

<p>Bodlaender et al. [7] proved a converse to Courcelle's Theorem for graphs [15] for the class of chordal graphs of bounded treewidth. Hliněný [25] generalised Courcelle's Theorem for graphs to classes of matroids represented over finite fields and of bounded branchwidth. This thesis has investigated the possibility of obtaining a generalisation of chordality to matroids that would enable us to prove a converse of Hliněný's Theorem [25]. There is a variety of equivalent characterisations for chordality in graphs. We have investigated the relationship between their generalisations to matroids. We prove that they are equivalent for binary matroids but typically inequivalent for more general classes of matroids. Supersolvability is a well studied property of matroids and, indeed, a graphic matroid is supersolvable if and only if its underlying graph is chordal. This is among the stronger ways of generalising chordality to matroids. However, to obtain the structural results that we need we require a stronger property that we call supersolvably saturated. Chordal graphs are well known to induce canonical tree decompositions. We show that supersolvably saturated matroids have the same property. These tree decompositions of supersolvably saturated matroids can be processed by a finite state automaton. However, they can not be completely described in monadic second-order logic. In order to express the matroids and their tree decompositions in monadic second-order logic we need to extend the logic over an extension field for each matroid represented over a finite field. We then use the fact that each maximal round modular flat of the tree decomposition for every matroid represented over a finite field, and in the specified class, spans a point in the vector space over the extension field. This enables us to derive a partial converse to Hliněný's Theorem.</p>

Chordality in Matroids: In Search of the Converse to Hliněný's Theorem

<p>Bodlaender et al. [7] proved a converse to Courcelle's Theorem for graphs [15] for the class of chordal graphs of bounded treewidth. Hliněný [25] generalised Courcelle's Theorem for graphs to classes of matroids represented over finite fields and of bounded branchwidth. This thesis has investigated the possibility of obtaining a generalisation of chordality to matroids that would enable us to prove a converse of Hliněný's Theorem [25]. There is a variety of equivalent characterisations for chordality in graphs. We have investigated the relationship between their generalisations to matroids. We prove that they are equivalent for binary matroids but typically inequivalent for more general classes of matroids. Supersolvability is a well studied property of matroids and, indeed, a graphic matroid is supersolvable if and only if its underlying graph is chordal. This is among the stronger ways of generalising chordality to matroids. However, to obtain the structural results that we need we require a stronger property that we call supersolvably saturated. Chordal graphs are well known to induce canonical tree decompositions. We show that supersolvably saturated matroids have the same property. These tree decompositions of supersolvably saturated matroids can be processed by a finite state automaton. However, they can not be completely described in monadic second-order logic. In order to express the matroids and their tree decompositions in monadic second-order logic we need to extend the logic over an extension field for each matroid represented over a finite field. We then use the fact that each maximal round modular flat of the tree decomposition for every matroid represented over a finite field, and in the specified class, spans a point in the vector space over the extension field. This enables us to derive a partial converse to Hliněný's Theorem.</p>

Inductive graph invariants and approximation algorithms

We introduce and study an inductively defined analogue [Formula: see text] of any increasing graph invariant [Formula: see text]. An invariant [Formula: see text] is increasing if [Formula: see text] whenever [Formula: see text] is an induced subgraph of [Formula: see text]. This inductive analogue simultaneously generalizes and unifies known notions like degeneracy, inductive independence number, etc., into a single generic notion. For any given increasing [Formula: see text], this gets us several new invariants and many of which are also increasing. It is also shown that [Formula: see text] is the minimum (over all orderings) of a value associated with each ordering. We also explore the possibility of computing [Formula: see text] (and a corresponding optimal vertex ordering) and identify some pairs [Formula: see text] for which [Formula: see text] can be computed efficiently for members of [Formula: see text]. In particular, it includes graphs of bounded [Formula: see text] values. Some specific examples (like the class of chordal graphs) have already been studied extensively. We further extend this new notion by (i) allowing vertex weighted graphs, (ii) allowing [Formula: see text] to take values from a totally ordered universe with a minimum and (iii) allowing the consideration of [Formula: see text]-neighborhoods for arbitrary but fixed [Formula: see text]. Such a generalization is employed in designing efficient approximations of some graph optimization problems. Precisely, we obtain efficient algorithms (by generalizing the known algorithm of Ye and Borodin [Y. Ye and A. Borodin, Elimination graphs, ACM Trans. Algorithms 8(2) (2012) 1–23] for special cases) for approximating optimal weighted induced [Formula: see text]-subgraphs and optimal [Formula: see text]-colorings (for hereditary [Formula: see text]’s) within multiplicative factors of (essentially) [Formula: see text] and [Formula: see text] respectively, where [Formula: see text] denotes the inductive analogue (as defined in this work) of optimal size of an unweighted induced [Formula: see text]-subgraph of the input and [Formula: see text] is the minimum size of a forbidden induced subgraph of [Formula: see text]. Our results generalize the previous result on efficiently approximating maximum independent sets and minimum colorings on graphs of bounded inductive independence number to optimal [Formula: see text]-subgraphs and [Formula: see text]-colorings for arbitrary hereditary classes [Formula: see text]. As a corollary, it is also shown that any maximal [Formula: see text]-subgraph approximates an optimal solution within a factor of [Formula: see text] for unweighted graphs, where [Formula: see text] is maximum size of any induced [Formula: see text]-subgraph in any local neighborhood [Formula: see text].

Total vertex-edge domination in graphs: Complexity and algorithms

Let [Formula: see text] be a simple, undirected and connected graph. A vertex [Formula: see text] of a simple, undirected graph [Formula: see text]-dominates all edges incident to at least one vertex in its closed neighborhood [Formula: see text]. A set [Formula: see text] of vertices is a vertex-edge dominating set of [Formula: see text], if every edge of graph [Formula: see text] is [Formula: see text]-dominated by some vertex of [Formula: see text]. A vertex-edge dominating set [Formula: see text] of [Formula: see text] is called a total vertex-edge dominating set if the induced subgraph [Formula: see text] has no isolated vertices. The total vertex-edge domination number [Formula: see text] is the minimum cardinality of a total vertex-edge dominating set of [Formula: see text]. In this paper, we prove that the decision problem corresponding to [Formula: see text] is NP-complete for chordal graphs, star convex bipartite graphs, comb convex bipartite graphs and planar graphs. The problem of determining [Formula: see text] of a graph [Formula: see text] is called the minimum total vertex-edge domination problem (MTVEDP). We prove that MTVEDP is linear time solvable for chain graphs and threshold graphs. We also show that MTVEDP can be approximated within approximation ratio of [Formula: see text]. It is shown that the domination and total vertex-edge domination problems are not equivalent in computational complexity aspects. Finally, an integer linear programming formulation for MTVEDP is presented.