Graphs Of Clique-width Research Articles

Some new algorithmic results on co-secure domination in graphs

For a simple graph G=(V,E) without any isolated vertex, a co-secure dominating set S of G satisfies two properties, (i) S is a dominating set of G, (ii) for every vertex v∈S, there exists a vertex u∈V∖S such that uv∈E and (S∖{v})∪{u} is a dominating set of G. The minimum cardinality of a co-secure dominating set of G is called the co-secure domination number of G and is denoted by γcs(G). The Co-secure Domination problem is to find a co-secure dominating set of a graph G of cardinality γcs(G). The decision version of the problem is known to be NP-complete for bipartite, planar, and split graphs. On the other hand, it is also known that the Co-secure Domination problem is efficiently solvable for proper interval graphs and cographs.In an effort to reduce the complexity gap of the Co-secure Domination problem, in this paper, we work on various important graph classes. We show that the decision version of the problem remains NP-complete for doubly chordal graphs and for some subclasses of bipartite graphs, namely, chordal bipartite graphs, star-convex bipartite graphs, and comb-convex bipartite graphs. On the positive side, we give an efficient algorithm to compute the co-secure domination number of chain graphs, which is an important subclass of bipartite graphs. In addition, we show that the problem is linear-time solvable for bounded tree-width graphs and bounded clique-width graphs. Further, we establish that the computational complexity of this problem differs from that of the classical domination problem.

On the computational difficulty of the terminal connection problem

A connection tree of a graph G for a terminal set W is a tree subgraph T of G such that leaves(T) ⊆ W ⊆ V(T). A non-terminal vertex is called linker if its degree in T is exactly 2, and it is called router if its degree in T is at least 3. The Terminal connection problem (TCP) asks whether G admits a connection tree for W with at most ℓ linkers and at most r routers, while the Steiner tree problem asks whether G admits a connection tree for W with at most k non-terminal vertices. We prove that, if r ≥ 1 is fixed, then TCP is polynomial-time solvable when restricted to split graphs. This result separates the complexity of TCP from the complexity of Steiner tree, which is known to be NP-complete on split graphs. Additionally, we prove that TCP is NP-complete on strongly chordal graphs, even if r ≥ 0 is fixed, whereas Steiner tree is known to be polynomial-time solvable. We also prove that, when parameterized by clique-width, TCP is W[1]-hard, whereas STeiner tree is known to be in FPT. On the other hand, agreeing with the complexity of Steiner tree, we prove that TCP is linear-time solvable when restricted to cographs (i.e. graphs of clique-width 2). Finally, we prove that, even if either ℓ ≥ 0 or r ≥ 0 is fixed, TCP remains NP-complete on graphs of maximum degree 3.

On general packing functions in graphs (Brief Announcement)

In this paper, we present the Generalized Packing Problem (GPF) in graphs in such a way that all other packing problems in graphs presented in the literature correspond to particular instances of it. We find the first class of graphs where one of these previously defined packing problems is tractable and another is NP-hard. We show that GPF remains linear-time solvable in strongly chordal graphs and, when reduced to instances with bounded neighborhood capacities (M-GPF), remains polynomial-time solvable in bounded clique-width graphs. To design specific algorithms that take advantage of the structural properties of certain subclasses of bounded clique-width graphs, we obtain technical results related to the behavior of the new packing parameter under several graph operations. In particular, we present a linear-time algorithm to solve GPF in block graphs and a polynomial-time algorithm to solve M-GPF in P4-tidy graphs.

Restrained condition on double Roman dominating functions

We continue the study of restrained double Roman domination in graphs. For a graph G=(V(G),E(G)), a double Roman dominating function f is called a restrained double Roman dominating function (RDRD function) if the subgraph induced by {v∈V(G)∣f(v)=0} has no isolated vertices. The restrained double Roman domination number (RDRD number) γrdR(G) is the minimum weight ∑v∈V(G)f(v) taken over all RDRD functions of G.We first prove that the problem of computing γrdR is NP-hard even for planar graphs, but it is solvable in linear time when restricted to bounded clique-width graphs such as trees, cographs and distance-hereditary graphs. Relationships between γrdR and some well-known parameters such as restrained domination number γr, domination number γ or restrained Roman domination number γrR are investigated in this paper by bounding γrdR from below and above involving γr, γ or γrR for general graphs, respectively. We prove that γrdR(T)≥n+2 for any tree T≠K1,n−1 of order n≥2 and characterize the family of all trees attaining the lower bound. The characterization of graphs with small RDRD numbers is given in this paper.

Optimal Centrality Computations Within Bounded Clique-Width Graphs

Given an n-vertex m-edge graph G of clique-width at most k, and a corresponding k-expression, we present algorithms for computing some well-known centrality indices (eccentricity and closeness) that run in \({\mathcal {O}}(2^{{\mathcal {O}}(k)}(n+m)^{1+\epsilon })\) time for any \(\epsilon > 0\). Doing so, we can solve various distance problems within the same amount of time, including: the diameter, the center, the Wiener index and the median set. Our run-times match conditional lower bounds of Coudert et al. (SODA’18) under the Strong Exponential-Time Hypothesis. On our way, we get a distance-labeling scheme for n-vertex m-edge graphs of clique-width at most k, using \({\mathcal {O}}(k\log ^2{n})\) bits per vertex and constructible in \(\tilde{\mathcal {O}}(k(n+m))\) time from a given k-expression. Doing so, we match the label size obtained by Courcelle and Vanicat (DAM 2016), while we considerably improve the dependency on k in their scheme. As a corollary, we get an \(\tilde{\mathcal {O}}(kn^2)\)-time algorithm for computing All-Pairs Shortest-Paths on n-vertex graphs of clique-width at most k, being given a k-expression. This partially answers an open question of Kratsch and Nelles (STACS’20). Our algorithms work for graphs with non-negative vertex-weights, under two different types of distances studied in the literature. For that, we introduce a new type of orthogonal range query as a side contribution of this work, that might be of independent interest.

Maximum Matching in Almost Linear Time on Graphs of Bounded Clique-Width

Recently, independent groups of researchers have presented algorithms to compute a maximum matching in \(\tilde{\mathcal{O}}(f(k) \cdot (n+m))\) time, for some computable function f, within the graphs where some clique-width upper bound is at most k (e.g., tree-width, modular-width and \(P_4\)-sparseness). However, to the best of our knowledge, the existence of such algorithm within the graphs of bounded clique-width has remained open until this paper. Indeed, we cannot even apply Courcelle’s theorem to this problem directly, because a matching cannot be expressed in \(MSO_1\) logic. Our first contribution is an almost linear-time algorithm to compute a maximum matching in any bounded clique-width graph, being given a corresponding clique-width expression. We also present how to compute the Edmonds-Gallai decomposition in almost linear time by using the same framework. For that, we do apply Courcelle’s theorem but to the classic Tutte-Berge formula, that can easily be expressed as a \(CMSO_1\) optimization problem. Doing so, we can compute the cardinality of a maximum matching, but not the matching itself. To obtain with this approach a maximum matching, we need to combine it with a recursive dissection scheme for bounded clique-width graphs and with a distributed version of Courcelle’s theorem (Courcelle and Vanicat, DAM 2016) – of which we present here a slightly stronger version than the standard one in the literature. Finally, for the bipartite graphs of clique-width at most k, we present an alternative \(\tilde{\mathcal{O}}(k^2\cdot (n+m))\)-time algorithm for the problem. The algorithm is randomized and it is based on a completely different approach than above: combining various reductions to matching and flow problems on bounded tree-width graphs with a very recent result on the parameterized complexity of linear programming (Dong et. al., STOC’21). Our results for bounded clique-width graphs extend many prior works on the complexity of Maximum Matching within cographs, distance-hereditary graphs, series-parallel graphs and other subclasses.

Maximal double Roman domination in graphs

A maximal double Roman dominating function (MDRDF) on a graph G=(V,E) is a function f:V(G)→{0,1,2,3} such that (i) every vertex v with f(v)=0 is adjacent to least two vertices assigned 2 or to at least one vertex assigned 3, (ii) every vertex v with f(v)=1 is adjacent to at least one vertex assigned 2 or 3 and (iii) the set {w∈V|f(w)=0} is not a dominating set of G. The weight of a MDRDF is the sum of its function values over all vertices, and the maximal double Roman domination number γdRm(G) is the minimum weight of an MDRDF on G. In this paper, we initiate the study of maximal double Roman domination. We first show that the problem of determining γdRm(G) is NP-complete for bipartite, chordal and planar graphs. But it is solvable in linear time for bounded clique-width graphs including trees, cographs and distance-hereditary graphs. Moreover, we establish various relationships relating γdRm(G) to some domination parameters. For the class of trees, we show that for every tree T of order n≥4,γdRm(T)≤54n and we characterize all trees attaining the bound. Finally, the exact values of γdRm(G) are given for paths and cycles.

Graphs of bounded cliquewidth are polynomially $χ$-bounded

We prove that if $\mathcal{C}$ is a hereditary class of graphs that is polynomially $\chi$-bounded, then the class of graphs that admit decompositions into pieces belonging to $\mathcal{C}$ along cuts of bounded rank is also polynomially $\chi$-bounded. In particular, this implies that for every positive integer $k$, the class of graphs of cliquewidth at most $k$ is polynomially $\chi$-bounded.

Fully Polynomial FPT Algorithms for Some Classes of Bounded Clique-width Graphs

Recently, hardness results for problems in P were achieved using reasonable complexity-theoretic assumptions such as the Strong Exponential Time Hypothesis. According to these assumptions, many graph-theoretic problems do not admit truly subquadratic algorithms. A central technique used to tackle the difficulty of the above-mentioned problems is fixed-parameter algorithms with polynomial dependency in the fixed parameter (P-FPT). Applying this technique to clique-width , an important graph parameter, remained to be done. In this article, we study several graph-theoretic problems for which hardness results exist such as cycle problems , distance problems , and maximum matching . We give hardness results and P-FPT algorithms, using clique-width and some of its upper bounds as parameters. We believe that our most important result is an algorithm in O ( k 4 ⋅ n + m )-time for computing a maximum matching, where k is either the modular-width of the graph or the P 4 -sparseness. The latter generalizes many algorithms that have been introduced so far for specific subclasses such as cographs. Our algorithms are based on preprocessing methods using modular decomposition and split decomposition. Thus they can also be generalized to some graph classes with unbounded clique-width.

Clique-width III

M AX -C UT , E DGE D OMINATING S ET , G RAPH C OLORING , and H AMILTONIAN C YCLE on graphs of bounded clique-width have received significant attention as they can be formulated in MSO 2 (and, therefore, have linear-time algorithms on bounded treewidth graphs by the celebrated Courcelle’s theorem), but cannot be formulated in MSO 1 (which would have yielded linear-time algorithms on bounded clique-width graphs by a well-known theorem of Courcelle, Makowsky, and Rotics). Each of these problems can be solved in time g ( k ) n f ( k ) on graphs of clique-width k . Fomin et al. (2010) showed that the running times cannot be improved to g ( k ) n O (1) assuming W[1]≠FPT. However, this does not rule out non-trivial improvements to the exponent f ( k ) in the running times. In a follow-up paper, Fomin et al. (2014) improved the running times for E DGE D OMINATING S ET and M AX -C UT to n O ( k ) , and proved that these problems cannot be solved in time g ( k ) n o ( k ) unless ETH fails. Thus, prior to this work, E DGE D OMINATING S ET and M AX -C UT were known to have tight n Θ ( k ) algorithmic upper and lower bounds. In this article, we provide lower bounds for H AMILTONIAN C YCLE and G RAPH C OLORING . For H AMILTONIAN C YCLE , our lower bound g ( k ) n o ( k ) matches asymptotically the recent upper bound n O ( k ) due to Bergougnoux, Kanté, and Kwon (2017). As opposed to the asymptotically tight n Θ( k ) bounds for E DGE D OMINATING S ET , M AX -C UT , and H AMILTONIAN C YCLE , the G RAPH C OLORING problem has an upper bound of n O (2 k ) and a lower bound of merely n o (√ [4] k ) (implicit from the W[1]-hardness proof). In this article, we close the gap for G RAPH C OLORING by proving a lower bound of n 2 o ( k ) . This shows that G RAPH C OLORING behaves qualitatively different from the other three problems. To the best of our knowledge, G RAPH C OLORING is the first natural problem known to require exponential dependence on the parameter in the exponent of n .

Degree-Constrained Orientation of Maximum Satisfaction: Graph Classes and Parameterized Complexity

The problem Max W-Light (Max W-Heavy) for an undirected graph is to assign a direction to each edge so that the number of vertices of outdegree at most W (resp. at least W) is maximized. It is known that these problems are NP-hard even for fixed W. For example, Max 0-Light is equivalent to the problem of finding a maximum independent set. In this paper, we show that for any fixed constant W, Max W-Heavy can be solved in linear time for hereditary graph classes for which treewidth is bounded by a function of degeneracy. We show that such graph classes include chordal graphs, circular-arc graphs, d-trapezoid graphs, chordal bipartite graphs, and graphs of bounded clique-width. To have a polynomial-time algorithm for Max W-Light, we need an additional condition of a polynomial upper bound on the number of potential maximal cliques to apply the metatheorem by Fomin et al. (SIAM J Comput 44:54---87, 2015). The aforementioned graph classes, except bounded clique-width graphs, satisfy such a condition. For graphs of bounded clique-width, we present a dynamic programming approach not using the metatheorem to show that it is actually polynomial-time solvable for this graph class too. We also study the parameterized complexity of the problems and show some tractability and intractability results.

Efficient computation of the characteristic polynomial of a threshold graph

An efficient algorithm is presented to compute the characteristic polynomial of a threshold graph. Threshold graphs were introduced by Chvátal and Hammer, as well as by Henderson and Zalcstein in 1977. A threshold graph is obtained from a one vertex graph by repeatedly adding either an isolated vertex or a dominating vertex, which is a vertex adjacent to all the other vertices. Threshold graphs are special kinds of cographs, which themselves are special kinds of graphs of clique-width 2. We obtain a running time of O(nlog2⁡n) for computing the characteristic polynomial, while the previously fastest algorithm ran in quadratic time. We improve the running time drastically in the case where there is a small number of alternations between 0's and 1's in the sequence defining a threshold graph.

Polynomial-time recognition of clique-width ≤3 graphs

Clique-width is a relatively new parameterization of graphs, philosophically similar to treewidth. Clique-width is more encompassing in the sense that a graph of bounded treewidth is also of bounded clique-width (but not the converse). For graphs of bounded clique-width, given the clique-width decomposition, every optimization, enumeration or evaluation problem that can be defined by a monadic second-order logic formula using quantifiers on vertices, but not on edges, can be solved in polynomial time.This is reminiscent of the situation for graphs of bounded treewidth, where the same statement holds even if quantifiers are also allowed on edges. Thus, graphs of bounded clique-width are a larger class than graphs of bounded treewidth, on which we can resolve fewer, but still many, optimization problems efficiently.One of the major open questions regarding clique-width is whether graphs of clique-width at most k, for fixed k, can be recognized in polynomial time. In this paper, we present the first polynomial-time algorithm (O(n2m)) to recognize graphs of clique-width at most 3.

Vertex disjoint paths on clique-width bounded graphs

We show that the l vertex disjoint paths problem between l pairs of vertices can be solved in linear time for co-graphs but is NP-complete for graphs of clique-width at most 6 and NLC-width at most 4. The NP-completeness follows from the fact that the line graph of a graph of tree-width k has clique-width at most 2 k + 2 and NLC-width at most k + 2 , and a result by Nishizeki et al. [The edge-disjoint paths problem is NP-complete for series-parallel graphs, Discrete Appl. Math. 115 (2001) 177–186]. The vertex disjoint paths problem is the first graph problem shown to be NP-complete on graphs of bounded clique-width but solvable in linear time on co-graphs and graphs of bounded tree-width. Additionally, we show that the r vertex disjoint paths problem between each of l pairs of vertices can be solved in polynomial time for co-graphs, if l is given to the input, and for graphs of bounded clique-width, if l is fixed.

Improved bottleneck domination algorithms

W.C.K. Yen introduced BOTTLENECK DOMINATION and BOTTLENECK INDEPENDENT DOMINATION. He presented an O ( n log n + m ) -time algorithm to compute a minimum bottleneck dominating set. He also obtained that the BOTTLENECK INDEPENDENT DOMINATING SET problem is NP-complete, even when restricted to planar graphs. We present simple linear time algorithms for the BOTTLENECK DOMINATING SET and the BOTTLENECK TOTAL DOMINATING SET problem. Furthermore, we give polynomial time algorithms (most of them with linear time-complexities) for the BOTTLENECK INDEPENDENT DOMINATING SET problem on the following graph classes: AT-free graphs, chordal graphs, split graphs, permutation graphs, graphs of bounded treewidth, and graphs of clique-width at most k with a given k-expression.

Approximating clique-width and branch-width

We construct a polynomial-time algorithm to approximate the branch-width of certain symmetric submodular functions, and give two applications. The first is to graph “clique-width.” Clique-width is a measure of the difficulty of decomposing a graph in a kind of tree-structure, and if a graph has clique-width at most k then the corresponding decomposition of the graph is called a “ k-expression.” We find (for fixed k) an O ( n 9 log n ) -time algorithm that, with input an n-vertex graph, outputs either a ( 2 3 k + 2 − 1 ) -expression for the graph, or a witness that the graph has clique-width at least k + 1 . (The best earlier algorithm, by Johansson [Ö. Johansson, log n -approximative NLC k -decomposition in O ( n 2 k + 1 ) time (extended abstract), in: Graph-Theoretic Concepts in Computer Science, Boltenhagen, 2001, in: Lecture Notes in Comput. Sci., vol. 2204, Springer, Berlin, 2001, pp. 229–240], constructs a 2 k log n -expression for graphs of clique-width at most k.) It was already known that several graph problems, NP-hard on general graphs, are solvable in polynomial time if the input graph comes equipped with a k-expression (for fixed k). As a consequence of our algorithm, the same conclusion follows under the weaker hypothesis that the input graph has clique-width at most k (thus, we no longer need to be provided with an explicit k-expression). Another application is to the area of matroid branch-width. For fixed k, we find an O ( n 3.5 ) -time algorithm that, with input an n-element matroid in terms of its rank oracle, either outputs a branch-decomposition of width at most 3 k − 1 or a witness that the matroid has branch-width at least k + 1 . The previous algorithm by Hliněný [P. Hliněný, A parametrized algorithm for matroid branch-width, SIAM J. Comput. 35 (2) (2005) 259–277] works only for matroids represented over a finite field.

GEM- AND CO-GEM-FREE GRAPHS HAVE BOUNDED CLIQUE-WIDTH

The P4 is the induced path of four vertices. The gem consists of a P4 with an additional universal vertex being completely adjacent to the P4, and the co-gem is its complement graph. Gem- and co-gem-free graphs generalize the popular class of cographs (i. e. P4-free graphs). The tree structure and algebraic generation of cographs has been crucial for several concepts of graph decomposition such as modular and homogeneous decomposition. Moreover, it is fundamental for the recently introduced concept of clique-width of graphs which extends the famous concept of treewidth. It is well-known that the cographs are exactly those graphs of clique-width at most 2. In this paper, we show that the clique-width of gem- and co-gem-free graphs is at most 16.

Linear Time Solvable Optimization Problems on Graphs of Bounded Clique-Width

Hierarchical decompositions of graphs are interesting for algorithmic purposes. There are several types of hierarchical decompositions. Tree decompositions are the best known ones. On graphs of tree-width at most k , i.e., that have tree decompositions of width at most k , where k is fixed, every decision or optimization problem expressible in monadic second-order logic has a linear algorithm. We prove that this is also the case for graphs of clique-width at most k , where this complexity measure is associated with hierarchical decompositions of another type, and where logical formulas are no longer allowed to use edge set quantifications. We develop applications to several classes of graphs that include cographs and are, like cographs, defined by forbidding subgraphs with ``too many'' induced paths with four vertices.

ON THE CLIQUE–WIDTH OF GRAPH WITH FEW P4'S

Babel and Olariu (1995) introduced the class of (q, t) graphs in which every set of q vertices has at most t distinct induced P4s. Graphs of clique-width at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be defined by k-expressions based on graph operations which use k vertex labels. In this paper we study the clique–width of the (q, t) graphs, for almost all possible combinations of q and t. On one hand we show that every (q, q - 3) graph for q ≥ 7, has clique–width ≤ q and a q–expression defining it can be obtained in linear time. On the other hand we show that the class of (q, q - 3) graphs for 4 ≤ q ≤ 6 and the class of (q, q - 1) graphs for q ≥ 4 are not of bounded clique-width.