# Subdomination in Graphs with Upper-Bounded Vertex Degree

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

## 3. Related Work

## 4. Results

**Definition 1.**

**Lemma 1.**

**Proof.**

- All variables are non-negative, so$${n}_{A}\ge 0,\dots ,{n}_{D}\ge 0,{n}_{AB}\ge 0,{n}_{AC}\ge 0,{n}_{BC}\ge 0,{n}_{CC}\ge 0.$$
- The degree of any vertex is bounded by $\Delta $, so the total number of edges that connect the vertices of class A or class C with vertices outside the class cannot exceed the number of vertices in the class multiplied by the degree upper bound $\Delta $ (diminished by 1 to account for loops):$${n}_{AB}+{n}_{AC}\le (\Delta -1){n}_{A},$$$${n}_{AC}+{n}_{BC}+{n}_{CC}\le (\Delta -1){n}_{C}.$$
- Vertices of class B have degree 3, so$${n}_{AB}+{n}_{BC}=2{n}_{B}.$$
- Vertices of class A vote “against”, so$${n}_{AB}\ge {n}_{AC}+2{n}_{A}.$$
- Vertices of class C vote “for”, so$${n}_{AC}+{n}_{CC}\ge {n}_{BC}.$$
- Graph G has n vertices in total, so$${n}_{A}+{n}_{B}+{n}_{C}\le n.$$
- Voting is positive, so$${n}_{B}+{n}_{C}\ge k.$$

**Theorem 1.**

**Proof.**

**Theorem 2.**

**Proof.**

**Theorem 3.**

**Proof.**

**Theorem 4.**

**Proof.**

**$\mathit{G}$ contains no unhappy vertices of degree 2.**Otherwise, let us consider an unhappy vertex v of degree 2 in G. This vertex votes “against” regardless of the labeling of its neighbor u, and the edge $uv$ can be removed without worsening the voting conditions for all vertices and without changing the number of unhappy vertices.**Let $\mathit{G}$ contain an unhappy vertex $\mathit{v}$ of degree 3. Then both of its neighbors (call them ${\mathit{u}}_{\mathbf{1}}$ and ${\mathit{u}}_{\mathbf{2}}$) are happy.**Otherwise, v votes “against”, and one can remove the edges ${u}_{1}v$ and ${u}_{2}v$ without worsening the voting conditions for v, ${u}_{1}$, and ${u}_{2}$ and reducing the number of edges, which contradicts the fact that G is an optimal graph with the minimum number of edges.**Let $\mathit{G}$ contain a happy vertex $\mathit{v}$ of degree 3. Then one of its neighbors (call it ${\mathit{u}}_{\mathbf{1}}$), is happy and the second neighbor (name it ${\mathit{u}}_{\mathbf{2}}$) is unhappy.**- (a)
- Otherwise, if ${u}_{1}$ and ${u}_{2}$ are happy, we can connect them directly with edge ${u}_{1}{u}_{2}$ and isolate vertex v. The votes of all vertices do not change and the number of edges decreases, which contradicts the fact that G is an optimal graph with the minimum number of edges.
- (b)
- If both ${u}_{1}$ and ${u}_{2}$ are unhappy, then, from Item 1 above, the degrees of ${u}_{1}$ and ${u}_{2}$ equal 3. Let us denote their second neighbors by ${w}_{1}$ and ${w}_{2}$ and consider all possible alternatives:
- i.
- If ${w}_{1}$ is happy, we can isolate vertices ${u}_{1}$ and v and directly connect vertices ${u}_{2}$ and ${w}_{1}$. In this case, the voting conditions for vertices ${u}_{2}$ and ${w}_{1}$ do not change. Vertex v in graph G votes “against” because it has two unhappy neighbors but it votes “for” after being isolated. Vertex ${u}_{1}$, on the contrary, votes “for” in G and votes “against” after being isolated. Hence, the total number of positive votes has not changed and opinion function f is still approving for the new graph, which is optimal since the number of happy vertices is the same. However, the number of components has increased by two, which contradicts the definition of G.
- ii.
- Now let both ${w}_{1}$ and ${w}_{2}$ be unhappy. Then vertices ${u}_{1}$, ${u}_{2}$, and v vote “against”, and the opinion of vertex v can be changed to $-1$ without changing the vote, which strictly reduces the number of happy vertices and contradicts the fact that G is optimal.
- iii.
- Finally, let ${w}_{1}$ be unhappy and ${w}_{2}$ be happy. Then, similarly to Case 3(b)i, we connect vertices ${u}_{1}$ and ${w}_{2}$ and isolate vertices ${u}_{2}$ and v increasing the number of components, which contradicts the definition of G.

**Graph $\mathit{G}$ has no vertices of degree 2.**We have already proved that no unhappy vertices of degree 2 exist. Let G have a happy vertex v of degree 2. This means that v is the beginning of a chain and, therefore, this chain has an end u, and, as follows from the above proof, u is also happy. A neighbor ${v}^{\prime}$ of v cannot be happy, otherwise one can add an edge $u{v}^{\prime}$ and isolate v leaving the opinion function approving over this new optimal graph and increasing the number of connected components by 1. If vertices ${u}^{\prime}$ and ${v}^{\prime}$ are unhappy, then one can isolate v by connecting ${v}^{\prime}$ directly to u. At the same time, the voting conditions of vertex ${v}^{\prime}$ do not change; u votes “against” as before and the voting conditions of vertex v have not worsened. However, the number of components increases, which contradicts the definition of G.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The “sunflower” graph of order n with odd maximum vertex degree $\Delta $ and voting quota k divisible by $\frac{\Delta -1}{2}$.

**Figure 2.**The “alternating sunflower” graph of order n with even maximum vertex degree $\Delta $ and voting quota k divisible by $\Delta -1$.

**Figure 3.**The optimal graph when the nonstrict upper bound for the vertex degree $\Delta $ equals 3.

**Figure 4.**The lower bound for the minimum support ${h}_{min}^{k}\left({\mathcal{G}}_{n,\Delta}\right)$ as a function of the nonstrict upper bound for the vertex degree $\Delta $ and voting quota k for graphs order $n=100$. Levels of the lower bound are depicted and labeled with red. Combinations of the values of parameters where the lower bound is proved to be tight are depicted with blue dots for Theorem 2, green dots for Theorem 3, and black dots for Theorem 4.

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**MDPI and ACS Style**

Lemtyuzhnikova, D.; Chebotarev, P.; Goubko, M.; Kudinov, I.; Shushko, N.
Subdomination in Graphs with Upper-Bounded Vertex Degree. *Mathematics* **2023**, *11*, 2722.
https://doi.org/10.3390/math11122722

**AMA Style**

Lemtyuzhnikova D, Chebotarev P, Goubko M, Kudinov I, Shushko N.
Subdomination in Graphs with Upper-Bounded Vertex Degree. *Mathematics*. 2023; 11(12):2722.
https://doi.org/10.3390/math11122722

**Chicago/Turabian Style**

Lemtyuzhnikova, Darya, Pavel Chebotarev, Mikhail Goubko, Ilja Kudinov, and Nikita Shushko.
2023. "Subdomination in Graphs with Upper-Bounded Vertex Degree" *Mathematics* 11, no. 12: 2722.
https://doi.org/10.3390/math11122722