Discrete Appl Math Research Articles

Counting kernels in directed graphs with arbitrary orientations

A kernel of a directed graph is a subset of vertices that is both independent and absorbing (every vertex not in the kernel has an out-neighbour in the kernel).Not all directed graphs contain kernels, and computing a kernel or deciding that none exist is NP-complete even on low-degree planar digraphs. The existing polynomial-time algorithms for this problem are all obtained by restricting both the undirected structure and the edge orientations of the input: for example, to chordal graphs without bidirectional edges (Pass-Lanneau, Igarashi and Meunier, Discrete Appl Math 2020) or to permutation graphs where each clique has a sink (Abbas and Saoula, 4OR 2005).By contrast, we count the kernels of a fuzzy circular interval graph in polynomial time, regardless of its edge orientations, and return a kernel when one exists. (Fuzzy circular interval graphs were introduced by Chudnovsky and Seymour in their structure theorem for claw-free graphs.)We also consider kernels on cographs, where we establish NP-hardness in general but linear-time solvability on the subclass of threshold graphs.

A Cubic Vertex-Kernel for Trivially Perfect Editing

We consider the Trivially Perfect Editing problem, where one is given an undirected graph \(G = (V,E)\) and a parameter \(k \in {\mathbb {N}}\) and seeks to edit (add or delete) at most k edges from G to obtain a trivially perfect graph. The related Trivially Perfect Completion and Trivially Perfect Deletion problems are obtained by only allowing edge additions or edge deletions, respectively. Trivially perfect graphs are both chordal and cographs, and have applications related to the tree-depth width parameter and to social network analysis. All variants of the problem are known to be NP-complete (Burzyn et al., in Discret Appl Math 154(13):1824–1844, 2006; Nastos and Gao, in Soc Netw 35(3):439–450, 2013) and to admit so-called polynomial kernels (Drange and Pilipczuk, in Algorithmica 80(12):3481–3524, 2018; Guo, in: Tokuyama, (ed) Algorithms and Computation, 18th International Symposium, ISAAC. Lecture Notes in Computer Science, Springer, Sendai, 2007. https://doi.org/10.1007/978-3-540-77120-3_79; Bathie et al., in Algorithmica 1–27, 2022). More precisely, Drange and Pilipczuk (Algorithmica 80(12):3481–3524, 2018) provided \(O(k^7)\) vertex-kernels for these problems and left open the existence of cubic vertex-kernels. In this work, we answer positively to this question for all three variants of the problem. Notice that a quadratic vertex-kernel was recently obtained for Trivially Perfect Completion by Bathie et al. (Algorithmica 1–27, 2022).

On Weak Flexibility in Planar Graphs

Recently, Dvořák, Norin, and Postle introduced flexibility as an extension of list coloring on graphs (J Graph Theory 92(3):191–206, 2019, https://doi.org/10.1002/jgt.22447). In this new setting, each vertex v in some subset of V(G) has a request for a certain color r(v) in its list of colors L(v). The goal is to find an L coloring satisfying many, but not necessarily all, of the requests. The main studied question is whether there exists a universal constant varepsilon >0 such that any graph G in some graph class mathscr {C} satisfies at least varepsilon proportion of the requests. More formally, for k > 0 the goal is to prove that for any graph G in mathscr {C} on vertex set V, with any list assignment L of size k for each vertex, and for every R subseteq V and a request vector (r(v): vin R, ~r(v) in L(v)), there exists an L-coloring of G satisfying at least varepsilon |R| requests. If this is true, then mathscr {C} is called varepsilon-flexible for lists of size k. Choi, Clemen, Ferrara, Horn, Ma, and Masařík (Discrete Appl Math 306:20–132, 2022, https://doi.org/10.1016/j.dam.2021.09.021) introduced the notion of weak flexibility, where R = V. We further develop this direction by introducing a tool to handle weak flexibility. We demonstrate this new tool by showing that for every positive integer b there exists varepsilon (b)>0 so that the class of planar graphs without K_4, C_5 , C_6 , C_7, B_b is weakly varepsilon (b)-flexible for lists of size 4 (here K_n, C_n and B_n are the complete graph, a cycle, and a book on n vertices, respectively). We also show that the class of planar graphs without K_4, C_5 , C_6 , C_7, B_5 is varepsilon-flexible for lists of size 4. The results are tight as these graph classes are not even 3-colorable.

Second-order productivity, second-order payoffs, and the Owen value

We introduce the concepts of the components’ second-order productivities in cooperative games with transferable utility (TU games) with a coalition structure (CS games) and of the components’ second-order payoffs for one-point solutions for CS games as generalizations of the players’ second-order productivities in TU games and of the players’ second-order payoffs for one-point solutions for TU games (Casajus in Discrete Appl Math 304:212–219, 2021). The players’ second-order productivities are conceptualized as second-order marginal contributions, that is, how one player affects another player’s marginal contributions to coalitions containing neither of them by entering these coalitions. The players’ second-order payoffs are conceptualized as the effect of one player leaving the game on the payoff of another player. Analogously, the components’ second-order productivities are conceptualized as their second-order productivities in the game between components; the components’ second-order payoffs are conceptualized as their second-order payoffs in the game between components. We show that the Owen value is the unique efficient one-point solution for CS games that reflects the players’ and the components’ second-order productivities in terms of their second-order payoffs.

Introducing lop-Kernels: A Framework for Kernelization Lower Bounds

In the Maximum Minimal Vertex Cover (MMVC) problem, we are given a graph G and a positive integer k, and the objective is to decide whether G contains a minimal vertex cover of size at least k. Motivated by the kernelization of MMVC with parameter k, our main contribution is to introduce a simple general framework to obtain kernelization lower bounds for a certain type of kernels for optimization problems, which we call lop -kernels. Informally, this type of kernel is required to preserve large optimal solutions in the reduced instance, and captures the vast majority of existing kernels in the literature. As a consequence of this framework, we show that the trivial quadratic kernel for MMVC is essentially optimal, answering a question of Boria et al. Discrete Appl Math 196:62–71, 2015. https://doi.org/10.1016/j.dam.2014.06.001), and that the known cubic kernel for Maximum Minimal Feedback Vertex Set is also essentially optimal. We present further applications for Tree Deletion Set and for Maximum Independent Set on \(K_t\)-free graphs. Back to the MMVC problem, given the (plausible) non-existence of subquadratic kernels for MMVC on general graphs, we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erdős–Hajnal property. Finally, we prove that MMVC does not admit polynomial kernels parameterized by the size of a minimum vertex cover of the input graph, even on bipartite graphs, unless \(\mathsf{NP} \subseteq \mathsf{coNP} / \mathsf{poly}\).

On the facets of stable set polytopes of circular interval graphs

A linear description of the stable set polytope STAB(G) of a quasi-line graph G is given in Eisenbrand et al. (Combinatorica 28(1):45–67, 2008), where the so called Ben Rebea Theorem (Oriolo in Discrete Appl Math 132(3):185–201, 2003) is proved. Such a theorem establishes that, for quasi-line graphs, STAB(G) is completely described by non-negativity constraints, clique inequalities, and clique family inequalities (CFIs). As quasi-line graphs are a superclass of line graphs, Ben Rebea Theorem can be seen as a generalization of Edmonds’ characterization of the matching polytope (Edmonds in J Res Natl Bureau Stand B 69:125–130, 1965), showing that the matching polytope can be described by non-negativity constraints, degree constraints and odd-set inequalities. Unfortunately, the description given by the Ben Rebea Theorem is not minimal, i.e., it is not known which are the (non-rank) clique family inequalities that are facet defining for STAB(G). To the contrary, it would be highly desirable to have a minimal description of STAB(G), pairing that of Edmonds and Pulleyblank (in: Berge, Chuadhuri (eds) Hypergraph seminar, pp 214–242, 1974) for the matching polytope. In this paper, we start the investigation of a minimal linear description for the stable set polytope of quasi-line graphs. We focus on circular interval graphs, a subclass of quasi-line graphs that is central in the proof of the Ben Rebea Theorem. For this class of graphs, we move an important step forward, showing some strong sufficient conditions for a CFI to induce a facet of STAB(G). In particular, such conditions come out to be related to the existence of certain proper circulant graphs as subgraphs of G. These results allows us to settle two conjectures on the structure of facet defining inequalities of the stable set polytope of circulant graphs (Pêcher and Wagler in Math Program 105:311–328, 2006) and of (fuzzy) circular graphs (Oriolo and Stauffer in Math Program 115:291–317, 2008), and to slightly refine the Ben Rebea Theorem itself.

An improvement of spectral conditions for Hamilton-connected graphs

A conjecture by Erds and Hajnal [Ramsey-type theorems. Discrete Appl Math. 1989;25:37–52] pointed out that the structure of graphs with forbidden induced subgraphs is very different from general graphs. More precisely, they contain much larger cliques or independent sets. Inspired by this, we prove that non-Hamilton-connected graphs satisfying certain conditions contain a large clique. From the perspective of looking for a maximum clique, we present some sufficient conditions for a graph to be Hamilton-connected in terms of size, spectral radius and signless Laplacian spectral radius, respectively, which extend some corresponding results.

The effect on A α -eigenvalues of mixed graphs and unit gain graphs by adding edges in clusters

Given a simple graph G, a -cluster of G is a pair of vertex subsets , where size of C is and every vertex in C shares the same set S of s neighbours. Let be a mixed graph whose underlying graph G contains a -cluster and let be a mixed graph on c vertices. Then is a graph obtained from by adding edges between some vertices in C such that . Let and be the adjacency matrix and the degree diagonal matrix of a graph G, respectively. Then the -matrix is defined to be , where . Assume that is a mixed graph containing a cognate -cluster , and for any two mixed graphs on c vertices satisfying each row sum of is equal to that of . We show that and share part of their -eigenvalues. Similarly, we consider the above problem on the unit gain graph. All of these results extend those of Cardoso and Rojo [Edge perturbation on graphs with clusters: adjacency, Laplacian and signless Laplacian eigenvalues. Linear Algebra Appl. 2017;512:113–128.] and those of Belardo, Brunetti and Ciampella [Edge perturbation on signed graphs with clusters: adjacency and Laplacian eigenvalues. Discrete Appl Math. 2019;269:130–138]. Based on our obtained results, we construct some pairs of -cospectral mixed graphs (with respect to two kinds of matrices for mixed graphs) which may not be switching equivalent.

On a Conjecture for Power Domination

The minimum cardinality of a power dominating set of a graph G is the power domination number of G, denoted by $$\gamma _P(G)$$ . We prove a conjecture on power domination posed by Benson et al. (Discrete Appl Math 251:103–113, 2018), which states that if G is a graph on n vertices such that every component of G and its complement $${\overline{G}}$$ have at least three vertices, then $$\gamma _P(G)+\gamma _P({\overline{G}})\le \lfloor \frac{n}{3}\rfloor +2$$ . Also, we show that if G is a graph on n vertices such that both G and $${\overline{G}}$$ are connected, then $$\gamma _P(G)+\gamma _P({\overline{G}})\le \lceil \frac{n}{3}\rceil +1$$ . This result improves a previous result due to Bensen et al.

Roman {k}-domination in trees and complexity results for some classes of graphs

In this paper, we study Roman {k}-dominating functions on a graph G with vertex set V for a positive integer k: a variant of {k}-dominating functions, generations of Roman $$\{2\}$$ -dominating functions and the characteristic functions of dominating sets, respectively, which unify classic domination parameters with certain Roman domination parameters on G. Let $$k\ge 1$$ be an integer, and a function $$f:V \rightarrow \{0,1,\dots ,k\}$$ defined on V called a Roman $$\{k\}$$ -dominating function if for every vertex $$v\in V$$ with $$f(v)=0$$ , $$\sum _{u\in N(v)}f(u)\ge k$$ , where N(v) is the open neighborhood of v in G. The minimum value $$\sum _{u\in V}f(u)$$ for a Roman $$\{k\}$$ -dominating function f on G is called the Roman $$\{k\}$$ -domination number of G, denoted by $$\gamma _{\{Rk\}}(G)$$ . We first present bounds on $$\gamma _{\{Rk\}}(G)$$ in terms of other domination parameters, including $$\gamma _{\{Rk\}}(G)\le k\gamma (G)$$ . Secondly, we show one of our main results: characterizing the trees achieving equality in the bound mentioned above, which generalizes M.A. Henning and W.F. klostermeyer’s results on the Roman {2}-domination number (Henning and Klostermeyer in Discrete Appl Math 217:557–564, 2017). Finally, we show that for every fixed $$k\in \mathbb {Z_{+}}$$ , associated decision problem for the Roman $$\{k\}$$ -domination is NP-complete, even for bipartite planar graphs, chordal bipartite graphs and undirected path graphs.

Global exact optimization for covering a rectangle with 6 circles

We address the problem of covering a rectangle with six identical circles, whose radius is to be minimized. We focus on open cases from Melissen and Schuur (Discrete Appl Math 99:149–156, 2000). Depending on the rectangle side lengths, different configurations of the circles, corresponding to the different ways they are placed, yield the optimal covering. We prove the optimality of the two configurations corresponding to open cases. For the first one, we propose a mathematical mixed-integer nonlinear optimization formulation, that allows one to compute global optimal solutions. For the second one, we provide an analytical expression of the optimal radius as a function of one of the rectangle side lengths. All open cases are thus closed for the optimal covering of a rectangle with six circles.

Independent Sets and Hitting Sets of Bicolored Rectangular Families

A bicolored rectangular family BRF is the collection of all axis-parallel rectangles formed by selecting a bottom-left corner from a finite set of points A and an upper-right corner from a finite set of points B. We devise a combinatorial algorithm to compute the maximum independent set and the minimum hitting set of a BRF that runs in \(O(n^{2.5}\sqrt{\log n})\)-time, where \(n=|A |+|B |\). This result significantly reduces the gap between the \(\Omega (n^7)\)-time algorithm by Benczúr (Discrete Appl Math 129 (2–3):233–262, 2003) for the more general problem of finding directed covers of pairs of sets, and the \(O(n^2)\)-time algorithms of Franzblau and Kleitman (Inf Control 63(3):164–189, 1984) and Knuth (ACM J Exp Algorithm 1:1, 1996) for BRFs where the points of A lie on an anti-diagonal line. Furthermore, when the bicolored rectangular family is weighted, we show that the problem of finding the maximum weight of an independent set is \(\mathbf {NP}\)-hard, and provide efficient algorithms to solve it on important subclasses.

Italian domination in the Cartesian product of paths

In a graph $$G=(V,E)$$ , each vertex $$v\in V$$ is assigned 0, 1 or 2 such that each vertex assigned 0 is adjacent to at least one vertex assigned 2 or two vertices assigned 1. Such an assignment is called an Italian dominating function (IDF) of G. The weight of an IDF f is $$w(f)=\sum _{v\in V}f(v)$$ . The Italian domination number of G is $$\gamma _{I}(G)=\min _{f} w(f)$$ . In this paper, we investigate the Italian domination number of the Cartesian product of paths, $$P_n\Box P_m$$ . We obtain the exact values of $$\gamma _{I}(P_n\Box P_2)$$ and $$\gamma _{I}(P_n\Box P_3)$$ . Also, we present a bound of $$\gamma _{I}(P_n\Box P_m)$$ for $$m\ge 4$$ , that is $$\frac{mn}{3}+\frac{m+n-4}{9}\le \gamma _{I}(P_{n}\Box P_{m})\le \frac{mn+2m+2n-8}{3}$$ where the lower bound is improved since the general lower bound is $$\frac{mn}{3}$$ presented by Chellali et al. (Discrete Appl Math 204:22–28, 2016). By the results of this paper, together with existing results, we give $$P_n\Box P_2$$ and $$P_n\Box P_3$$ are examples for which $$\gamma _{I}=\gamma _{r2}$$ where $$\gamma _{r2}$$ is the 2-rainbow domination number. This can partially solve the open problem presented by Bresar et al. (Discrete Appl Math 155:2394–2400, 2007). Finally, Vizing’s conjecture on Italian domination in $$P_n\Box P_m$$ is checked.

Upper Bounds on the k-Tuple (Roman) Domination Number of a Graph

Rautenbach and Volkmann (Appl Math Lett 20:98–102, 2007), gave an upper bound for the k-tuple domination number of a graph. Rad (J Combin Math Comb Comput, 2019, in press) presented an improvement of the above bound using the Caro-Wei Theorem. In this paper, using the well-known Brooks’ Theorem for vertex coloring and vertex covers, we improve the above bounds on the k-tuple domination number under some certain conditions. In the special case $$k=1$$ , we improve the upper bounds for the domination number (Arnautov in Prikl Mat Program 11:3–8, 1974; Payan in Cahiers Centre Etudes Recherche Oper 17:307–317, 1975) and the Roman domination number (Cockayne et al. in Discrete Math 278:11–22, 2004). We also improve bounds given by Hansberg and Volkmann (Discrete Appl Math 157:1634–1639, 2009) for Roman k-domination number, and Rad and Rahbani (Discuss Math Graph Theory 39:41–53, 2019) for double Roman domination number.

On the existence and non-existence of some classes of bent–negabent functions

In this paper we investigate different questions related to bent–negabent functions. We first take an expository look at the state-of-the-art research in this domain and point out some technical flaws in certain results and fix some of them. Further, we derive a necessary and sufficient condition for which the functions of the form $${\mathbf{x}}\cdot \pi ({\mathbf{y}})\oplus h({\mathbf{y}})$$ [Maiorana–McFarland ( $${\mathcal {M}}$$ )] is bent–negabent, and more generally, we study the non-existence of bent–negabent functions in the $${\mathcal {M}}$$ class. We also identify some functions that are bent–negabent. Next, we continue the recent work by Mandal et al. (Discrete Appl Math 236:1–6, 2018) on rotation symmetric bent–negabent functions and show their non-existence in larger classes. For example, we prove that there is no rotation symmetric bent–negabent function in $$4p^k$$ variables, where p is an odd prime. We present the non-existence of such functions in certain classes that are affine transformations of rotation symmetric functions. Keeping in mind the existing literature, we correct here some technical issues and errors found in other papers and provide some novel results.

The minimal-ABC trees with $$B_2$$-branches

The atom–bond connectivity (ABC) index is a degree-based graph topological index with a lot of chemical applications, including those of predicting the stability of alkanes and the strain energy of cycloalkanes. It is known (Chen and Guo in MATCH Commun Math Comput Chem 65:713–722, 2011; Das et al. in MATCH Commun Math Comput Chem 76:159–170, 2011) that among all connected graphs, trees minimize the ABC index (such trees are called minimal-ABC trees). Several structural properties of minimal-ABC trees were proved in the past several years. Here we continue to make a step forward towards the complete characterization of the minimal-ABC trees. In Dimitrov (Discrete Appl Math 204:90–116, 2016), it was shown that a minimal-ABC tree cannot contain more than 11 so-called $$B_2$$-branches. We improve this result by showing that if a minimal-ABC tree of order larger than 39 contains so-called $$B_1$$-branches, then it contains exactly one $$B_2$$-branch, and if a minimal-ABC tree of order larger than 163 contains no $$B_1$$-branch, then it contains at most two $$B_2$$-branches.

On reduced second Zagreb index

The reduced second Zagreb index $$RM_2$$ of a graph G is defined as $$RM_2(G)=\sum _{uv\in E(G)}(d_G(u)-1)(d_G(v)-1)$$, where $$d_G(u)$$ is the degree of the vertex u of graph G. Furtula et al. (Discrete Appl Math 178: 83–88, 2014) studied the difference between the classical Zagreb indices of graphs and showed that it is closely related to $$RM_2$$. In this paper, we obtain an upper bound in terms of order n and size m on $$RM_2$$ of $$K_{r+1}$$-free graphs. Also we prove that among all graphs of order n with chromatic number $$\chi $$, the Turan graph $$T_{n,\,\chi }$$ is the unique graph having the maximum $$RM_2$$. Furthermore, we completely characterize the extremal graphs with respect to $$RM_2$$ among all unicyclic graphs of order n with girth g.

Graphs with equal domination and covering numbers

A dominating set of a graph G is a set Dsubseteq V_G such that every vertex in V_G-D is adjacent to at least one vertex in D, and the domination number gamma (G) of G is the minimum cardinality of a dominating set of G. A set Csubseteq V_G is a covering set of G if every edge of G has at least one vertex in C. The covering number beta (G) of G is the minimum cardinality of a covering set of G. The set of connected graphs G for which gamma (G)=beta (G) is denoted by {mathcal {C}}_{gamma =beta }, whereas {mathcal {B}} denotes the set of all connected bipartite graphs in which the domination number is equal to the cardinality of the smaller partite set. In this paper, we provide alternative characterizations of graphs belonging to {mathcal {C}}_{gamma =beta } and {mathcal {B}}. Next, we present a quadratic time algorithm for recognizing bipartite graphs belonging to {mathcal {B}}, and, as a side result, we conclude that the algorithm of Arumugam et al. (Discrete Appl Math 161:1859–1867, 2013) allows to recognize all the graphs belonging to the set {mathcal {C}}_{gamma =beta } in quadratic time either. Finally, we consider the related problem of patrolling grids with mobile guards, and show that it can be solved in O(n log n + m) time, where n is the number of line segments of the input grid and m is the number of its intersection points.

Using contracted solution graphs for solving reconfiguration problems

We introduce in a general setting a dynamic programming method for solving reconfiguration problems. Our method is based on contracted solution graphs, which are obtained from solution graphs by performing an appropriate series of edge contractions that decrease the graph size without losing any critical information needed to solve the reconfiguration problem under consideration. Our general framework captures the approach behind known reconfiguration results of Bonsma (Discrete Appl Math 231:95–112, 2017) and Hatanaka et al. (IEICE Trans Fundam Electron Commun Comput Sci 98(6):1168–1178, 2015). As a third example, we apply the method to the following well-studied problem: given two k-colorings $$\alpha $$ and $$\beta $$ of a graph G, can $$\alpha $$ be modified into $$\beta $$ by recoloring one vertex of G at a time, while maintaining a k-coloring throughout? This problem is known to be PSPACE-hard even for bipartite planar graphs and $$k=4$$ . By applying our method in combination with a thorough exploitation of the graph structure we obtain a polynomial-time algorithm for $$(k-2)$$ -connected chordal graphs.

A further construction of asymptotically optimal codebooks with multiplicative characters

Codebooks with small inner-product correlation are preferred in many practical applications such as direct spread code division multiple access communications, coding theory, compressed sensing and so on. Heng et al. (IEEE Trans Inf Theory 63(10):6179–6187, 2017), Heng (Discrete Appl Math 250:227–240, 2018) and Luo and Cao (IEEE Trans Inf Theory 64(10):6498–6505, 2018) proposed constructions of asymptotically optimal codebooks via character sums related to Jacobi sums. In this paper, we mainly present a further generalization of the work by Heng (2018). The construction in this paper is an extension and generalization of the above works, and deduces a class of asymptotically optimal codebooks. Furthermore, the parameters of the codebooks are flexible and new.