arXiv subjects

Tınaz Ekim

Publications and source records attributed to Tınaz Ekim.

At least 19 recordsLinked to original sources

arXiv2026

Perfect Graph Modification Problems: An Integer Programming Approach

Graph modification problems aim to find a small set of modifications to a graph so that it satisfies a desired property. The literature is rather rich in NP-completeness results and polynomial time solvable cases for special graph classes. However, no exact algorithm has been proposed for perfect graph modification problems. In this work, we propose the first exact solution methods based on integer programming for three variants: minimum perfect editing, minimum perfect completion, and the perfect sandwich problems. The minimum perfect editing problem inquires about the smallest number of edge additions and deletions needed to make a graph perfect, while the completion problem allows only for edge additions. The perfect sandwich problem is a decision problem that asks whether a perfect graph can be formed by adding edges from a restricted subset. To solve these problems, we formulate an integer programming model based on the Strong Perfect Graph Theorem. To address the resulting exponential number of constraints, we propose a branch-and-cut algorithm that dynamically generates them on demand. At the core of this approach is an efficient separation routine for enumerating odd holes and odd antiholes. We also release this underlying routine as "is_perfect"-a standalone open-source perfect graph recognizer and odd hole enumerator designed for broader community reuse. To enhance the practical efficiency of the branch-and-cut algorithm, we calculate the expected number of odd holes and odd antiholes in random Erdos Renyi graphs. In addition, we propose "IterativeModificationHeuristic", the first heuristic for the editing and completion problems, which provides upper bounds. Finally, we demonstrate the empirical effectiveness of the proposed methods through computational experiments on a wide range of instance types; all benchmark instances are publicly available.

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arXiv2026

Totally Disjoint Diametral Paths

In this paper, we study totally disjoint diametral paths in simple connected graphs. A diametral path in a graph is a shortest path that connects two vertices whose mutual distance is equal to the diameter of the graph. Totally disjoint paths are paths that have no vertices in common, including their end vertices. We show that the problem of deciding whether a graph $G$ has $k$ totally disjoint diametral paths is NP-complete. We consider restricted classes of graphs for which the problem of determining the maximum size of a set of totally disjoint diametral paths is readily solved. We then give a linear-time algorithm for a subclass of maximal outerplanar graphs called 2-paths, define a polynomial-time algorithm for threshold graphs, and establish a structural bound for proper interval graphs. Finally, we define classes of extremal graphs with $k$ totally disjoint diametral paths of length $d$ having the fewest possible number of edges.

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arXiv2026

A Benders Decomposition Approach for the k-Defensive Domination Problem

The k-defensive domination problem is a powerful modeling tool for strategic decision-making in network security and disaster/emergency management, where multiple nodes may be simultaneously under attack. Despite its practical relevance, the problem has been poorly studied, largely due to its high computational difficulty. This study investigates the application of Benders decomposition to the k-defensive domination problem, aiming to improve computational efficiency over standard integer programming formulations. Several cut generation strategies, including a combinatorial approach and the simultaneous addition of multiple cuts, are proposed. Theoretical results on the strength of feasibility cuts are presented. In addition, two novel enhancement strategies are proposed: a clique-cover-based heuristic, the first feasible solution method in the literature for this problem, achieving up to 98% improvement over the trivial upper bound, and an initial cut generation heuristic that in some cases resolves the problem without further branching. Experiments on Erdos-Renyi, chordal, and Barabasi-Albert instances show that the algorithm variant involving all of the proposed components is able to solve instances that remain out of reach for classical formulations and standard Benders decomposition.

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arXiv2025

Sparse Sets in Triangle-free Graphs

A set of vertices is $k$-sparse if it induces a graph with a maximum degree of at most $k$. In this missive, we consider the order of the largest $k$-sparse set in a triangle-free graph of fixed order. We show, for example, that every triangle-free graph of order 11 contains a 1-sparse 5-set; every triangle-free graph of order 13 contains a 2-sparse 7-set; and every triangle-free graph of order 8 contains a 3-sparse 6-set. Further, these are all best possible. For fixed $k$, we consider the growth rate of the largest $k$-sparse set of a triangle-free graph of order $n$. Also, we consider Ramsey numbers of the following type. Given $i$, what is the smallest $n$ having the property that all triangle-free graphs of order $n$ contain a 4-cycle or a $k$-sparse set of order $i$. We use both direct proof techniques and an efficient graph enumeration algorithm to obtain several values for defective Ramsey numbers and a parameter related to largest sparse sets in triangle-free graphs, along with their extremal graphs.

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arXiv2024

Generation of weighted trees, block trees and block graphs

We present a general framework to generate trees every vertex of which has a non-negative weight and a color. The colors are used to impose certain restrictions on the weight and colors of other vertices. We first extend the enumeration algorithms of unweighted trees given in [19, 20] to generate weighted trees that allow zero weight. We avoid isomorphisms by generalizing the concept of centroids to weighted trees and then using the so-called centroid-rooted canonical weighted trees. We provide a time complexity analysis of unranking algorithms and also show that the output delay complexity of enumeration is linear. The framework can be used to generate graph classes taking advantage of their tree-based decompositions/representations. We demonstrate our framework by generating weighted block trees which are in one-to-one correspondence with connected block graphs. All connected block graphs up to 19 vertices are publicly available at [1].

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arXiv2023

Constructing extremal triangle-free graphs using integer programming

The maximum number of edges in a graph with matching number m and maximum degree d has been determined in [1] and [2], where some extremal graphs have also been provided. Then, a new question has emerged: how the maximum edge count is affected by forbidding some subgraphs occurring in these extremal graphs? In [3], the problem is solved in triangle-free graphs for $d \geq m$, and for $d < m$ with either $Z(d) \leq m < 2d$ or $d \leq 6$, where $Z(d)$ is approximately $5d/4$. The authors derived structural properties of triangle-free extremal graphs, which allows us to focus on constructing small extremal components to form an extremal graph. Based on these findings, in this paper, we develop an integer programming formulation for constructing extremal graphs. Since our formulation is highly symmetric, we use our own implementation of Orbital Branching to reduce symmetry. We also implement our integer programming formulation so that the feasible region is restricted iteratively. Using a combination of the two approaches, we expand the solution into $d \leq 10$ instead of $d \leq 6$ for $m > d$. Our results endorse the formula for the number of edges in all extremal triangle-free graphs conjectured in [3].

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arXiv2022

Maximum size of a triangle-free graph with bounded maximum degree and matching number

Determining the maximum number of edges under degree and matching number constraints have been solved for general graphs by Chvátal and Hanson (1976), and by Balachandran and Khare (2009). It follows from the structure of those extremal graphs that deciding whether this maximum number decreases or not when restricted to claw-free graphs, to $C_4$-free graphs or to triangle-free graphs are separately interesting research questions. The first two cases being already settled, respectively by Dibek, Ekim and Heggernes (2017), and by Blair, Heggernes, Lima and D.Lokshtanov (2020). In this paper we focus on triangle-free graphs. We show that unlike most cases for claw-free graphs and $C_4$-free graphs, forbidding triangles from extremal graphs causes a strict decrease in the number of edges and adds to the hardness of the problem. We provide a formula giving the maximum number of edges in a triangle-free graph with degree at most $d$ and matching number at most $m$ for all cases where $d\geq m$, and for the cases where $d<m$ with either $d\leq 6$ or $Z(d)\leq m < 2d$ where $Z(d)$ is a function of $d$ which is roughly $5d/4$. We also provide an integer programming formulation for the remaining cases and as a result of further discussion on this formulation, we conjecture that our formula giving the size of triangle-free extremal graphs is also valid for these open cases.

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arXiv2022

Critical Equimatchable Graphs

A graph G is equimatchable if every maximal matching of G has the same cardinality. In this paper, we investigate equimatchable graphs such that the removal of any edge harms the equimatchability, called edge-critical equimatchable graphs (ECE-graphs). We show that apart from two simple cases, namely bipartite ECE-graphs and even cliques, all ECE-graphs are 2-connected factor-critical. Accordingly, we give a characterization of factor-critical ECE-graphs with connectivity 2. Our result provides a partial answer to an open question posed by Levit and Mandrescu on the characterization of well-covered graphs with no shedding vertex. We also introduce equimatchable graphs such that the removal of any vertex harms the equimatchability, called vertex-critical equimatchable graphs (VCE-graphs). To conclude, we enlighten the relationship between various subclasses of equimatchable graphs (including ECE-graphs and VCE-graphs) and discuss the properties of factor-critical ECE-graphs with connectivity at least 3.

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arXiv2021

Defective Ramsey Numbers and Defective Cocolorings in Some Subclasses of Perfect Graphs

In this paper, we investigate a variant of Ramsey numbers called defective Ramsey numbers where cliques and independent sets are generalized to $k$-dense and $k$-sparse sets, both commonly called $k$-defective sets. We focus on the computation of defective Ramsey numbers restricted to some subclasses of perfect graphs. Since direct proof techniques are often insufficient for obtaining new values of defective Ramsey numbers, we provide a generic algorithm to compute defective Ramsey numbers in a given target graph class. We combine direct proof techniques with our efficient graph generation algorithm to compute several new defective Ramsey numbers in perfect graphs, bipartite graphs and chordal graphs. We also initiate the study of a related parameter, denoted by $c^{\mathcal G}_k(m)$, which is the maximum order $n$ such that the vertex set of any graph of order at $n$ in a class $\mathcal{G}$ can be partitioned into at most $m$ subsets each of which is $k$-defective. We obtain several values for $c^{\mathcal G}_k(m)$ in perfect graphs and cographs.

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arXiv2020

An Exact Cutting Plane Algorithm to Solve the Selective Graph Coloring Problem in Perfect Graphs

We consider the selective graph coloring problem, which is a generalization of the classical graph coloring problem. Given a graph together with a partition of its vertex set into clusters, we want to choose exactly one vertex per cluster so that the number of colors needed to color the selected set of vertices is minimized. This problem is known to be NP-hard. In this study, we focus on an exact cutting plane algorithm for selective graph coloring in perfect graphs. Since there exists no suite of perfect graph instances to the best of our knowledge, we also propose an algorithm to randomly (but not uniformly) generate perfect graphs, and provide a large collection of instances available online. We conduct computational experiments to test our method on graphs with varying size and densities, and compare our results with a state-of-the-art algorithm from the literature and with solving an integer programming formulation of the problem by CPLEX. Our experiments demonstrate that our solution strategy significantly improves the solvability of the problem.

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arXiv2020

Defensive Domination in Proper Interval Graphs

$k$-defensive domination, a variant of the classical domination problem on graphs, seeks a minimum cardinality vertex set providing a surjective defense against any attack on vertices of cardinality bounded by a parameter $k$. The problem has been shown to be NP-complete} for fixed $k$; if $k$ is part of the input, the problem is not even in NP. We present efficient algorithms solving this problem on proper interval graphs with $k$ part of the input. The algorithms take advantage of the linear orderings of the end points of the intervals associated with vertices to realize a greedy approach to solution. The first algorithm is based on the interval model and has complexity ${\cal O}(n \cdot k)$ for a graph on $n$ vertices. The second one is an improvement of the first and employs bubble representations of proper interval graph to realize an improved complexity of ${\cal O}(n+ \vert{\cal B}\vert \cdot \log k)$ for a graph represented by $\vert{\cal B}\vert$ bubbles.

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arXiv2020

Well-Totally-Dominated Graphs

A subset of vertices in a graph is called a total dominating set if every vertex of the graph is adjacent to at least one vertex of this set. A total dominating set is called minimal if it does not properly contain another total dominating set. In this paper, we study graphs whose all minimal total dominating sets have the same size, referred to as well-totally-dominated (WTD) graphs. We first show that WTD graphs with bounded total domination number can be recognized in polynomial time. Then we focus on WTD graphs with total domination number two. In this case, we characterize triangle-free WTD graphs and WTD graphs with packing number two, and we show that there are only finitely many planar WTD graphs with minimum degree at least three. Lastly, we show that if the minimum degree is at least three then the girth of a WTD graph is at most 12. We conclude with several open questions.

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arXiv2020

On the Maximum Cardinality Cut Problem in Proper Interval Graphs and Related Graph Classes

Although it has been claimed in two different papers that the maximum cardinality cut problem is polynomial-time solvable for proper interval graphs, both of them turned out to be erroneous. In this paper, we give FPT algorithms for the maximum cardinality cut problem in classes of graphs containing proper interval graphs and mixed unit interval graphs when parameterized by some new parameters that we introduce. These new parameters are related to a generalization of the so-called bubble representations of proper interval graphs and mixed unit interval graphs and to clique-width decompositions.

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arXiv2019

Defective Ramsey Numbers in Graph Classes

Given a graph $G$, a $k$-sparse $j$-set is a set of $j$ vertices inducing a subgraph with maximum degree at most $k$. A $k$-dense $i$-set is a set of $i$ vertices that is $k$-sparse in the complement of $G$. As a generalization of Ramsey numbers, the $k$-defective Ramsey number $R_k^{\mathcal{G}}(i,j)$ for the graph class $\mathcal{G}$ is defined as the smallest natural number $n$ such that all graphs on $n$ vertices in the class $\mathcal{G}$ have either a $k$-dense $i$-set or a $k$-sparse $j$-set. In this paper, we examine $R_k^{\mathcal{G}}(i,j)$ where $\mathcal{G}$ represents various graph classes. In forests and cographs, we give formulas for all defective Ramsey numbers. In cacti, bipartite graphs and split graphs, we provide defective Ramsey numbers in most of the cases and point out open questions, formulated as conjectures if possible.

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arXiv2019

The Complexity of Subtree Intersection Representation of Chordal Graphs and Linear Time Chordal Graph Generation

It is known that any chordal graph on $n$ vertices can be represented as the intersection of $n$ subtrees in a tree on $n$ nodes. This fact is recently used in [2] to generate random chordal graphs on $n$ vertices by generating $n$ subtrees of a tree on $n$ nodes. It follows that the space (and thus time) complexity of such an algorithm is at least the sum of the sizes of the generated subtrees assuming that a tree is given by a set of nodes. In [2], this complexity was mistakenly claimed to be linear in the number $m$ of edges of the generated chordal graph. This error is corrected in [3] where the space complexity is shown to be $Ω(m n^{1/4})$. The exact complexity of the algorithm is left as an open question. In this paper, we show that the sum of the sizes of $n$ subtrees in a tree on $n$ nodes is $Θ(m\sqrt{n})$. We also show that we can confine ourselves to contraction-minimal subtree intersection representations since they are sufficient to generate every chordal graph. Furthermore, the sum of the sizes of the subtrees in such a representation is at most $2m+n$. We use this result to derive the first linear time random chordal graph generator. Based on contraction-minimal representations, we also derive structural properties of chordal graphs related to their connectivity. In addition to these theoretical results, we conduct experiments to study the quality of the chordal graphs generated by our algorithm and compare them to those in the literature. Our experimental study indicates that the generated graphs do not have a restricted structure and the sizes of maximal cliques are distributed fairly over the range. Furthermore, our algorithm is simple to implement and produces graphs with 10000 vertices and $4 . 10^7$ edges in less than one second on a laptop computer.

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arXiv2018

Mind the Independence Gap

The independence gap of a graph was introduced by Ekim et al. (2018) as a measure of how far a graph is from being well-covered. It is defined as the difference between the maximum and minimum size of a maximal independent set. We investigate the independence gap of a graph from structural and algorithmic points of view, with a focus on classes of perfect graphs. Generalizing results on well-covered graphs due to Dean and Zito (1994) and Hujdurović et al. (2018), we express the independence gap of a perfect graph in terms of clique partitions and use this characterization to develop a polynomial-time algorithm for recognizing graphs of constant independence gap in any class of perfect graphs of bounded clique number. Next, we introduce a hereditary variant of the parameter, which we call hereditary independence gap and which measures the maximum independence gap over all induced subgraphs of the graph. We show that determining whether a given graph has hereditary independence gap at most $k$ is polynomial-time solvable if $k$ is fixed and co-NP-complete if $k$ is part of input. We also investigate the complexity of the independent set problem in graph classes related to independence gap, showing that the problem is NP-complete in the class of graphs of independence gap at most one and polynomial-time solvable in any class of graphs with bounded hereditary independence gap. Combined with some known results on claw-free graphs, our results imply that the independent domination problem is solvable in polynomial time in the class of $\{$claw, 2$P_3\}$-free graphs.

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arXiv2018

On Almost Well-Covered Graphs of Girth at Least 6

We consider a relaxation of the concept of well-covered graphs, which are graphs with all maximal independent sets of the same size. The extent to which a graph fails to be well-covered can be measured by its independence gap, defined as the difference between the maximum and minimum sizes of a maximal independent set in $G$. While the well-covered graphs are exactly the graphs of independence gap zero, we investigate in this paper graphs of independence gap one, which we also call almost well-covered graphs. Previous works due to Finbow et al. (1994) and Barbosa et al. (2013) have implications for the structure of almost well-covered graphs of girth at least $k$ for $k\in \{7,8\}$. We focus on almost well-covered graphs of girth at least $6$. We show that every graph in this class has at most two vertices each of which is adjacent to exactly $2$ leaves. We give efficiently testable characterizations of almost well-covered graphs of girth at least $6$ having exactly one or exactly two such vertices. Building on these results, we develop a polynomial-time recognition algorithm of almost well-covered $\{C_3,C_4,C_5,C_7\}$-free graphs.

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arXiv2018

Generation of random chordal graphs using subtrees of a tree

Chordal graphs form one of the most studied graph classes. Several graph problems that are NP-hard in general become solvable in polynomial time on chordal graphs, whereas many others remain NP-hard. For a large group of problems among the latter, approximation algorithms, parameterized algorithms, and algorithms with moderately exponential or sub-exponential running time have been designed. Chordal graphs have also gained increasing interest during the recent years in the area of enumeration algorithms. Being able to test these algorithms on instances of chordal graphs is crucial for understanding the concepts of tractability of hard problems on graph classes. Unfortunately, only few studies give algorithms for generating chordal graphs. Even in these papers, only very few methods aim for generating a large variety of chordal graphs. Surprisingly, none of these methods is directly based on the "intersection of subtrees of a tree" characterization of chordal graphs. In this paper, we give an algorithm for generating chordal graphs, based on the characterization that a graph is chordal if and only if it is the intersection graph of subtrees of a tree. Upon generating a random host tree, we give and test various methods that generate subtrees of the host tree. We compare our methods to one another and to existing ones for generating chordal graphs. Our experiments show that one of our methods is able to generate the largest variety of chordal graphs in terms of maximal clique sizes. Moreover, two of our subtree generation methods result in an overall complexity of our generation algorithm that is the best possible time complexity for a method generating the entire node set of subtrees in a "intersection of subtrees of a tree" representation. The instances corresponding to the results presented in this paper, and also a set of relatively small-sized instances are made available online.

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