arXiv subjects

Andrey A. Dobrynin

Publications and source records attributed to Andrey A. Dobrynin.

6 recordsLinked to original sources

arXiv2025

On restrained coalitions in graphs: bounds and exact values

A subset $D \subseteq V$ is a dominating set of a graph $G$ with vertex set $V$ if every vertex $v \in V \setminus D$ is adjacent to a vertex in $D$. Two subsets of $V$ form a coalition if neither of them is a dominating set, but their union is a dominating set. A coalition partition of $G$ is its vertex partition $π$ such that every non-dominating set of $π$ is a member of some coalition, and every dominating set is a single-vertex set in $π$. The coalition number $C(G)$ of a graph $G$ is the maximum cardinality of its coalition partitions. A subset $R \subseteq V$ is a restrained dominating set if $R$ is a dominating set and any vertex of $V \setminus R$ has at least one neighbor in $V \setminus R$. Restrained dominating coalition, restrained dominating partition and restrained coalition number $RC(G)$ are defined by the same way. In this paper, we prove that $RC(G) \le C(G)$ for an arbitrary graph $G$. In addition, the restrained coalition numbers of cycles and trees are determined.

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arXiv2025

Coalition graphs of connected domination partitions in subcubic graphs

A graph is subcubic if it is connected and its maximum vertex degree does not exceed 3. Two disjoint vertex subsets of a graph $G$ form a connected coalition in $G$ if neither of them is a connected dominating set but their union is a connected dominating set. A connected coalition partition of $G$ is a partition of its vertices $π(G) = \{V_1, V_2,..., V_k \}$, such that each $V_i$ is either a connected dominating set consisting of a single vertex or forms a coalition with some set of $π(G)$. The formation of connected coalitions is described by a coalition graph whose vertices correspond to the sets of $π$, and two vertices are adjacent if and only if the corresponding sets form a coalition in $G$. We characterize all coalition graphs of subcubic graphs.

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arXiv2024

On cubic graphs having the maximal coalition number

A coalition in a graph $G$ with vertex set $V$ consists of two disjoint sets $V_1, V_2\subset V$ such that neither $V_1$ nor $V_2$ is a dominating set, but the union $V_1\cup V_2$ is a dominating set in $G$. A partition of graph vertices is called a coalition partition $\mathcal{P}$ if every non-dominating set of $\mathcal{P}$ is a member of a coalition and every dominating set is a single-vertex set. The coalition number $C(G)$ of a graph $G$ is the maximum cardinality of its coalition partition. It is known that for cubic graphs $C(G)\le 9$. The existence of cubic graphs with the maximal coalition number is an unsolved problem. In this paper, an infinite family of cubic graphs satisfying $C(G)=9$ is constructed.

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arXiv2023

Some results on the Wiener index related to the Šoltés problem of graphs

The Wiener index, $W(G)$, of a connected graph $G$ is the sum of distances between its vertices. In 2021, Akhmejanova et al. posed the problem of finding graphs $G$ with large $R_m(G)= |\{v\in V(G)\,|\,W(G)-W(G-v)=m \in \mathbb{Z} \}|/ |V(G)|$. It is shown that there is a graph $G$ with $R_m(G) > 1/2$ for any integer $m \ge 0$. In particular, there is a regular graph of even degree with this property for any odd $m \ge 1$. The proposed approach allows to construct new families of graphs $G$ with $R_0(G) \rightarrow 1/2$ when the order of $G$ increases.

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arXiv2021

On the Wiener (r,s)-complexity of fullerene graphs

Fullerene graphs are mathematical models of fullerene molecules. The Wiener $(r,s)$-complexity of a fullerene graph $G$ with vertex set $V(G)$ is the number of pairwise distinct values of $(r,s)$-transmission $tr_{r,s}(v)$ of its vertices $v$: $tr_{r,s}(v)= \sum_{u \in V(G)} \sum_{i=r}^{s} d(v,u)^i$ for positive integer $r$ and $s$. The Wiener $(1,1)$-complexity is known as the Wiener complexity of a graph. Irregular graphs have maximum complexity equal to the number of vertices. No irregular fullerene graphs are known for the Wiener complexity. Fullerene (IPR fullerene) graphs with n vertices having the maximal Wiener $(r,s)$-complexity are counted for all $n\le 100$ ($n\le 136$) and small $r$ and $s$. The irregular fullerene graphs are also presented.

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arXiv2019

On the Wiener complexity and the Wiener index of fullerene graphs

Fullerenes are molecules in the form of cage-like polyhedra, consisting solely of carbon atoms. Fullerene graphs are mathematical models of fullerene molecules. The transmission of a vertex $v$ of a graph is the sum of distances from $v$ to all the other vertices. The number of different vertex transmissions is called the Wiener complexity of a graph. Some calculation results on the Wiener complexity and the Wiener index of fullerene graphs of order $n \le 216$ are presented. Structure of graphs with the maximal Wiener complexity or the maximal Wiener index is discussed and formulas for the Wiener index of several families of graphs are obtained.

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