arXiv subjects

Yasemin Büyükçolak

Publications and source records attributed to Yasemin Büyükçolak.

4 recordsLinked to original sources

arXiv2026

Well-VE-Dominated Graphs

Vertex-edge domination is a natural variant of domination in which a vertex ve-dominates an edge whenever it is incident to the edge or adjacent to one of its endpoints. A set of vertices is a vedominating set if it ve-dominates every edge of the graph. In this paper, we introduce the notion of wellve- dominated graphs, namely graphs in which all minimal ve-dominating sets have the same cardinality. Equivalently, these are graphs whose minimal isolating sets all have the same cardinality; we refer to them as well-isolated graphs. We prove that recognizing well-ve-dominated graphs is co-NP-complete. We determine perfectly well-ve-dominated graphs, the largest hereditary subclass, and obtain both a complete forbidden induced subgraph characterization and a linear-time recognition algorithm. In particular, every connected nontrivial member of this subclass has vertex-edge domination and isolation number one. Finally, we characterize the class of well-ve-dominated trees. More precisely, we prove that well-ve-domination and well-ve-coveredness coincide on nontrivial trees, where well-ve-coveredness requires all minimal independent ve-dominating sets to have the same cardinality. The characterization yields a linear-time recognition algorithm and identifies the reduced members with the extremal trees attaining equality in the sharp isolation bound.

math.CO↗

arXiv2021

Minimum Reload Cost Cycle Cover in Complete Graphs

The reload cost refers to the cost that occurs along a path on an edge-colored graph when it traverses an internal vertex between two edges of different colors. Galbiati et al.[1] introduced the Minimum Reload Cost Cycle Cover problem, which is to find a set of vertex-disjoint cycles spanning all vertices with minimum reload cost. They proved that this problem is strongly NP-hard and not approximable within $1/ε$ for any $ε> 0$ even when the number of colors is 2, the reload costs are symmetric and satisfy the triangle inequality. In this paper, we study this problem in complete graphs having equitable or nearly equitable 2-edge-colorings, which are edge-colorings with two colors such that for each vertex $v \in V(G)$, $||c_1(v)| -|c_2(v)|| \leq 1$ or $||c_1(v)| -|c_2(v)|| \leq 2$, respectively, where $c_i(v)$ is the set of edges with color $i$ that is incident to $v$. We prove that except possibly on complete graphs with fewer than 13 vertices, the minimum reload cost is zero on complete graphs with nearly equitable 2-edge-colorings by proving the existence of a monochromatic cycle cover. Furthermore, we provide a polynomial-time algorithm that constructs a monochromatic cycle cover in complete graphs with an equitable 2-edge-coloring except possibly in a complete graph with four vertices. Our algorithm also finds a monochromatic cycle cover in complete graphs with a nearly equitable 2-edge-coloring except some special cases.

cs.DM↗

arXiv2021

Triangle-Free Equimatchable Graphs

A graph is called equimatchable if all of its maximal matchings have the same size. Frendrup et al. [8] provided a characterization of equimatchable graphs with girth at least $5$. In this paper, we extend this result by providing a complete structural characterization of equimatchable graphs with girth at least $4$, i.e., equimatchable graphs with no triangle, by identifying the equimatchable triangle-free graph families. Our characterization also extends the result given by Akbari et al. in [1], which proves that the only connected triangle-free equimatchable $r$-regular graphs are $C_5$, $C_7$ and $K_{r,r}$, where $r$ is a positive integer. Given a non-bipartite graph, our characterization implies a linear time recognition algorithm for triangle-free equimatchable graphs.

cs.DM↗

arXiv2016

A Simple Extension of Dirac's Theorem on Hamiltonicity

The classical Dirac theorem asserts that every graph $G$ on $n$ vertices with minimum degree $δ(G) \ge \lceil n/2 \rceil$ is Hamiltonian. The lower bound of $\lceil n/2 \rceil$ on the minimum degree of a graph is tight. In this paper, we extend the classical Dirac theorem to the case where $δ(G) \ge \lfloor n/2 \rfloor $ by identifying the only non-Hamiltonian graph families in this case. We first present a short and simple proof. We then provide an alternative proof that is constructive and self-contained. Consequently, we provide a polynomial-time algorithm that constructs a Hamiltonian cycle, if exists, of a graph $G$ with $δ(G) \ge \lfloor n/2 \rfloor$, or determines that the graph is non-Hamiltonian. Finally, we present a self-contained proof for our algorithm which provides insight into the structure of Hamiltonian cycles when $δ(G) \ge \lfloor n/2 \rfloor$ and is promising for extending the results of this paper to the cases with smaller degree bounds.

cs.DM↗