arXiv subjects

Hamidreza Golmohammadi

Publications and source records attributed to Hamidreza Golmohammadi.

7 recordsLinked to original sources

arXiv2026

Total outer-independent coalition in graphs

A set $D$ of vertices graph $G$ is a total outer-independent dominating set (TOIDS) of $G$ if every vertex of $G$ is adjacent to at least one vertex in $D$, and $V(G)\setminus D$ is an independent set of $G$. A TOI-coalition in $G$ comprises two disjoint sets of vertices $A$ and $B$ of $G$, neither of which is a TOIDS but whose union $A\cup B$ is a TOIDS of $G$. We say that the sets $A$ and $B$ form a TOI-coalition, and are TOI-coalition partners. A TOI-coalition partition in $G$ is a vertex partition $Ψ=\{V_1, V_2,..., V_k\}$ in which every set forms a TOI-coalition with another set in $Ψ$. The TOI-coalition number $C_t^{oi}(G)$ is the maximum cardinality among all TOI-coalition partitions of $G$. In this work, the above-mentioned concepts are introduced and studied. The existence of a TOI-coalition partition is investigated. Several sharp upper bounds on $C_t^{oi}(G)$ are established. Finally, the exact values of $C_t^{oi}(G)$ for some graph classes are obtained.

math.CO↗

arXiv2024

Introduction to double coalitions in graphs

Let $G(V, E)$ be a finite, simple, isolate-free graph. A set $D$ of vertices of a graph $G$ with the vertex set $V$ is a double dominating set of $G$, if every vertex $v\in D$ has at least one neighbor in $D$ and every vertex $v \in V \setminus D$ has at least two neighbors in $D$. A double coalition consists of two disjoint sets of vertices $V_{1}$ and $V_{2}$, neither of which is a double dominating set but their union $V_{1}\cup V_{2}$ is a double dominating set. A double coalition partition of a graph $G$ is a partition $Π= \{V_1, V_2,..., V_k \}$ of $V$ such that no subset of $Π$ is a double dominating set of $G$, but for every set $V_i \in Π$, there exists a set $V_j \in Π$ such that $V_i$ and $V_j$ form a double coalition. In this paper, we study properties of double coalitions in graphs.

math.CO↗

arXiv2024

Strong coalitions in graphs

For a graph $G=(V,E)$, a set $D\subset V(G)$ is a strong dominating set of $G$, if for every vertex $x\in V (G)\setminus D$ there is a vertex $y\in D$ with $xy \in E(G)$ and $deg(x)\leq deg(y)$. A strong coalition consists of two disjoint sets of vertices $V_{1}$ and $V_{2}$, neither of which is a strong dominating set but whose union $V_{1}\cup V_{2}$, is a strong dominating set. A vertex partition $Ω=\{V_1, V_2,..., V_k \}$ of vertices in $G$ is a strong coalition partition, if every set $V_i \inΩ$ either is a strong dominating set consisting of a single vertex of degree $n-1$, or is not a strong dominating set but produces a strong coalition with another set $V_j \in Ω$ that is not a strong dominating set. The maximum cardinality of a strong coalition partition of $G$ is the strong coalition number of $G$ and is denoted by $SC(G)$. In this paper, we study properties of strong coalitions in graphs.

math.CO↗

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.

math.CO↗

arXiv2023

Independent domination stability in graphs

A non-empty set $S\subseteq V (G)$ of the simple graph $G=(V(G),E(G))$ is an independent dominating set of $G$ if every vertex not in $S$ is adjacent with some vertex in $S$ and the vertices of $S$ are pairwise non-adjacent. The independent domination number of $G$, denoted by $γ_i(G)$, is the minimum size of all independent dominating sets of $G$. The independent domination stability, or simply $id$-stability of $G$ is the minimum number of vertices whose removal changes the independent domination number of $G$. In this paper, we investigate properties of independent domination stability in graphs. In particular, we obtain several bounds and obtain the independent domination stability of some operations of two graphs.

math.CO↗

arXiv2023

Connected coalitions in graphs

The connected coalition in a graph $G=(V,E)$ consists of two disjoint sets of vertices $V_{1}$ and $V_{2}$, neither of which is a connected dominating set but whose union $V_{1}\cup V_{2}$, is a connected dominating set. A connected coalition partition in a graph $G$ of order $n=|V|$ is a vertex partition $ψ$ = $\{V_1, V_2,..., V_k \}$ such that every set $V_i \in ψ$ either is a connected dominating set consisting of a single vertex of degree $n-1$, or is not a connected dominating set but forms a connected coalition with another set $V_j\in ψ$ which is not a connected dominating set. The connected coalition number, denoted by $CC(G)$, is the maximum cardinality of a connected coalition partition of $G$. In this paper, we initiate the study of connected coalition in graphs and present some basic results. Precisely, we characterize all graphs that have a connected coalition partition. Moreover, we show that for any graph $G$ of order $n$ with $δ(G)=1$ and with no full vertex, it holds that $CC(G)<n$. Furthermore, we show that for any tree $T$, $CC(T)=2$. Finally, we present two polynomial-time algorithms that for a given connected graph $G$ of order $n$ determine whether $CC(G)=n$ or $CC(G)=n-1$.

math.CO↗

arXiv2022

Introduction to total coalitions in graphs

Let $G$ be a graph with vertex set $V$. Two disjoint sets $V_1, V_2\subseteq V$ are called a total coalition in $G$, if neither $V_1$ and $V_2$ is a total dominating set of $G$ but $V_1\cup V_2$ is a total dominating set. A total coalition partition of $G$ is a vertex partition $π=\{V_1,V_2,\ldots, V_k\}$ such that no set of $π$ is a total dominating set but each set $V_i\in π$ forms a total coalition with another set $V_j\in π$. The maximum cardinality of a total coalition partition of $G$ is called the total coalition number of $G$, denoted by $TC(G)$. In this paper, we initiate the study of the total coalition in graphs and its properties.

math.CO↗