The Minimum Dominating Set problem is polynomial for (claw, P8)-free graphs

On the Erdős-Gyárfás conjecture in claw-free graphs

The Erdős-Gyárfás conjecture states that every graph with minimum degree at least three has a cycle whose length is a power of 2. Since this conjecture has proven to be far from reach, Hobbs asked if the Erdős-Gyárfás conjecture holds in claw-free graphs. In this paper, we obtain some results on this question, in particular for cubic claw-free graphs.

Maximum Number of Minimum Dominating and Minimum Total Dominating Sets

Given a connected graph with domination (or total domination) number γ>=2, we ask for the maximum number m_γand m_{γ,T} of dominating and total dominating sets of size γ. An exact answer is provided for γ=2and lower bounds are given for γ>=3.

关于固定直径图的支配集问题的复杂性

On the complexity of Dominating Set for graphs with fixed diameter

本文针对固定直径的无爪图研究支配集问题的复杂性，证明直径至少为3时该问题NP-完全，直径至多为2时多项式时间可解，并借助2K₂-自由图上的最小极大匹配算法完成证明。

唯一配对 vs 边顶最小支配集在树中的问题

Unique paired vs edge-vertex minimum dominating sets in trees

本文证明了树中唯一最小边顶支配集与唯一最小配对支配集的等价性。

关于森林中最小支配集与全支配集的数量

On the number of minimum dominating sets and total dominating sets in forests

本文研究森林中最小支配集与全支配集的数量，证明支配数为γ的森林最多有√5^γ个最小支配集，并构造下界接近该值的树，同时否定 Henning 等人关于最小全支配集数量的猜想。

A characterization of trees having a minimum vertex cover which is also a minimum total dominating set

A vertex cover of a graph $G = (V, E)$ is a set $X \subseteq V$ such that each edge of $G$ is incident to at least one vertex of $X$. A dominating set $D \subseteq V$ is a total dominating set of $G$ if the subgraph induced by $D$ has no isolated vertices. A $(γ_t-τ)$-set of $G$ is a minimum vertex cover which is also a minimum total dominating set. In this article we give a constructive characterization of trees having a $(γ_t-τ)$-set.

On the maximum number of minimum dominating sets in forests

Fricke, Hedetniemi, Hedetniemi, and Hutson asked whether every tree with domination number $γ$ has at most $2^γ$ minimum dominating sets. Bien gave a counterexample, which allows to construct forests with domination number $γ$ and $2.0598^γ$ minimum dominating sets. We show that every forest with domination number $γ$ has at most $2.4606^γ$ minimum dominating sets, and that every tree with independence number $α$ has at most $2^{α-1}+1$ maximum independent sets.

$1/2$-conjectures on the domination game and claw-free graphs

Let $γ_g(G)$ be the game domination number of a graph $G$. Rall conjectured that if $G$ is a traceable graph, then $γ_g(G) \le \left\lceil \frac{1}{2}n(G)\right\rceil$. Our main result verifies the conjecture over the class of line graphs. Moreover, in this paper we put forward the conjecture that if $δ(G) \geq 2$, then $γ_g(G) \leq \left\lceil \frac{1}{2}n(G) \right\rceil$. We show that both conjectures hold true for claw-free cubic graphs. We further prove the upper bound $γ_g(G) \le \left\lceil \frac{11}{20} \, n(G) \right\rceil$ over the class of claw-free graphs of minimum degree at least…

Extremal problems on the Hamiltonicity of claw-free graphs

In 1962, Erdős proved that if a graph $G$ with $n$ vertices satisfies $$ e(G)>\max\left\{\binom{n-k}{2}+k^2,\binom{\lceil(n+1)/2\rceil}{2}+\left\lfloor \frac{n-1}{2}\right\rfloor^2\right\}, $$ where the minimum degree $δ(G)\geq k$ and $1\leq k\leq(n-1)/2$, then it is Hamiltonian. For $n \geq 2k+1$, let $E^k_n=K_{k}\vee (kK_1+K_{n-2k})$, where "$\vee$" is the "join" operation. One can observe $e(E^k_n)=\binom{n-k}{2}+k^2$ and $E^k_n$ is not Hamiltonian. As $E^k_n$ contains induced claws for $k\geq 2$, a natural question is to characterize all 2-connected claw-free non-Hamiltonian graphs with th…

Vertices in all minimum paired-dominating sets of block graphs

Let $G=(V,E)$ be a simple graph without isolated vertices. A set $S\subseteq V$ is a paired-dominating set if every vertex in $V-S$ has at least one neighbor in $S$ and the subgraph induced by $S$ contains a perfect matching. In this paper, we present a linear-time algorithm to determine whether a given vertex in a block graph is contained in all its minimum paired-dominating sets.

立方图中的广义支配结构

Generalized domination structure in cubic graphs

本文研究了立方图中的广义支配结构，通过特定方法确定最小支配集，并在多项式时间内解决该问题。

关于真增强幂图为无爪图的有限群

On the finite group whose proper enhanced power graph is claw-free

本文研究真增强幂图为无爪图的有限群，证明增强幂图无爪等价于群为循环群，利用循环化子刻画商群结构，分别给出可解非幂零群与不可解群的分类结果。

无爪图的Schur正性猜想的一个反例

A counterexample to the claw-free Schur-positivity conjecture

该研究针对Stanley记载的无爪图Schur正性猜想，构造出12顶点的反例，经计算验证其色对称函数系数为负，且12顶点是最小反例阶数，还发现另一个同构类反例。

Boxicity and Cubicity of Asteroidal Triple free graphs

An axis parallel $d$-dimensional box is the Cartesian product $R_1 \times R_2 \times ... \times R_d$ where each $R_i$ is a closed interval on the real line. The {\it boxicity} of a graph $G$, denoted as $\boxi(G)$, is the minimum integer $d$ such that $G$ can be represented as the intersection graph of a collection of $d$-dimensional boxes. An axis parallel unit cube in $d$-dimensional space or a $d$-cube is defined as the Cartesian product $R_1 \times R_2 \times ... \times R_d$ where each $R_i$ is a closed interval on the real line of the form $[a_i,a_i + 1]$. The {\it cubicity} of $G$, denot…

The tight bound for the strong chromatic indices of claw-free subcubic graphs

Let $G$ be a graph and $k$ a positive integer. A strong $k$-edge-coloring of $G$ is a mapping $ϕ: E(G)\to \{1,2,\dots,k\}$ such that for any two edges $e$ and $e'$ that are either adjacent to each other or adjacent to a common edge, $ϕ(e)\neq ϕ(e')$. The strong chromatic index of $G$, denoted as $χ'_{s}(G)$, is the minimum integer $k$ such that $G$ has a strong $k$-edge-coloring. Lv, Li and Zhang [Graphs and Combinatorics 38 (3) (2022) 63] proved that if $G$ is a claw-free subcubic graph other than the triangular prism then $χ_s'(G)\le 8$. In addition, they asked if the upper bound $8$ can be …

Bounds on the Maximum Number of Minimum Dominating Sets

We use probabilistic methods to find lower bounds on the maximum number, in a graph with domination number γ, of dominating sets of size γ. We find that we can randomly generate a graph that, w.h.p., is dominated by almost all sets of size γ. At the same time, we use a modified adjacency matrix to obtain lower bounds on the number of sets of a given size that do not dominate a graph on n vertices

无爪立方图与零强制

Claw-free cubic graphs and zero forcing

本文针对无爪立方图零强制数上界的三个开放问题展开研究，刻画了满足特定条件的连通无爪立方图，还为具有哈密顿收缩多重图的无爪立方图建立了改进上界，涉及零强制数与图中三角形、菱形数量的关系。

无爪图中的度和条件与具有有界圈数的2-因子

Degree sum conditions and a 2-factor with a bounded number of cycles in claw-free graphs

本文利用无爪图的闭包概念，在独立集度和条件与 $σ_{k+1}(G)\geq n$ 条件下证明了无爪图存在至多含 $k$ 个圈的2-因子，并由此部分解决了 Faudree 等人2012年的猜想。

Blocking dominating sets for $H$-free graphs via edge contractions

In this paper, we consider the following problem: given a connected graph $G$, can we reduce the domination number of $G$ by one by using only one edge contraction? We show that the problem is $\mathsf{NP}$-hard when restricted to $\{P_6,P_4+P_2\}$-free graphs and that it is $\mathsf{coNP}$-hard when restricted to subcubic claw-free graphs and $2P_3$-free graphs. As a consequence, we are able to establish a complexity dichotomy for the problem on $H$-free graphs when $H$ is connected.

Loop zero forcing and grundy domination in planar graphs and claw-free cubic graphs

Given a simple, finite graph with vertex set $V(G)$, we define a zero forcing set of $G$ as follows. Choose $S\subseteq V(G)$ and color all vertices of $S$ blue and all vertices in $V(G) - S$ white. The color change rule is if $w$ is the only white neighbor of blue vertex $v$, then we change the color of $w$ from white to blue. If after applying the color change rule as many times as possible eventually every vertex of $G$ is blue, we call $S$ a zero forcing set of $G$. $Z(G)$ denotes the minimum cardinality of a zero forcing set. Davila and Henning proved in \cite{zerocubic} that for any claw…

The Game of Cops and Robber on (Claw, Even-hole)-free Graphs

In this paper, we study the game of cops and robber on the class of graphs with no even hole (induced cycle of even length) and claw (a star with three leaves). The cop number of a graph $G$ is defined as the minimum number of cops needed to capture the robber. Here, we prove that the cop number of all claw-free even-hole-free graphs is at most two and, in addition, the capture time is at most $2n$ rounds, where $n$ is the number of vertices of the graph. Moreover, our results can be viewed as a first step towards studying the structure of claw-free even-hole-free graphs.

Disjunctive domination in graphs with minimum degree at least two

A set $D$ of vertices in $G$ is a disjunctive dominating set in $G$ if every vertex not in $D$ is adjacent to a vertex of $D$ or has at least two vertices in $D$ at distance $2$ from it in $G$. The disjunctive domination number, $γ^{d}_2(G)$, of $G$ is the minimum cardinality of a disjunctive dominating set in $G$. In this paper, we show that if $G$ be a graph of order at least $3$, $δ(G)\geq 2$ and with no component isomorphic to any of eight forbidden graphs, then $γ^{d}_2(G)\leq \frac{|G|}{3}$. Moreover, we provide an infinite family of graphs attaining this bound. In addition, we also stud…

Total Forcing and Zero Forcing in Claw-Free Cubic Graphs

A dynamic coloring of the vertices of a graph $G$ starts with an initial subset $S$ of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set $S$ is called a forcing set (zero forcing set) of $G$ if, by iteratively applying the forcing process, every vertex in $G$ becomes colored. If the initial set $S$ has the added property that it induces a subgraph of $G$ without isolated vertices, then $S$ is called a total forcing set in $G$. The…

关于森林中最小全控制集的最大数量

On the maximum number of minimum total dominating sets in forests

本文提出关于树的最小全控制集数量的猜想，并证明森林中该数量的上界，包含指数和多项式形式的界。

On Perfect and Quasiperfect Domination in Graphs

A subset $S\subseteq V$ in a graph $G=(V,E)$ is a $k$-quasiperfect dominating set (for $k\geq 1$) if every vertex not in $S$ is adjacent to at least one and at most $k$ vertices in $S$. The cardinality of a minimum $k$-quasiperfect dominating set in $G$ is denoted by $γ_ {\stackrel{}{1k}}(G)$. Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept and allow us to construct a decreasing chain of quasiperfect dominating numbers $ n \ge γ_ {\stackrel{}{11}}(G) \ge γ_ {\stackrel{}{12}}(G)\ge \ldots \ge γ_ {\stackrel{}{1Δ}}(G)=γ(G)$ in order…

On the circumference, connectivity and dominating cycles

Every 4-connected graph with minimum degree $δ$ and connectivity $κ$ either has a cycle of length at least $4δ-2κ$ or has a dominating cycle.

Three domination number and connectivity in graphs

In a graph G, a vertex dominates itself and its neighbors. A subset S of V is called a dominating set in G if every vertex in V is dominated by at least one vertex in S. The domination number gamma G is the minimum cardinality of a dominating set. A set S subset V is called a double dominating set of a graph G if every vertex in V is dominated by at least two vertices in S. The minimum cardinality of a double dominating set is called double domination number of G. In a graph G, a vertex dominates itself and its neighbors. A subset S of V is called a dominating set in G if every vertex in V is …

Mycielski构造下的位置支配型问题

Location-Domination Type Problems Under the Mycielski Construction

本文研究了Mycielski构造下定位支配集等变体的最小基数问题，给出了一般上下界，并针对星图、路径和环给出了改进或达到已知上界的具体结果。

Super Domination: Graph Classes, Products and Enumeration

The dominating set problem (DSP) is one of the most famous problems in combinatorial optimization. It is defined as follows. For a given simple graph $G=(V,E)$, a dominating set of $G$ is a subset $S\subseteq V$ such that every vertex in $ V \setminus S$ is adjacent to at least one vertex in $S$. Furthermore, the DSP is the problem of finding a minimum-size dominating set and the corresponding minimum size, the domination number of $G$. In this, work we investigate a variant of the DSP, the super dominating set problem (SDSP), which has attracted much attention during the last years. A dominat…

New results on metric-locating-dominating sets of graphs

A dominating set $S$ of a graph is a metric-locating-dominating set if each vertex of the graph is uniquely distinguished by its distances from the elements of $S$, and the minimum cardinality of such a set is called the metric-location-domination number. In this paper, we undertake a study that, in general graphs and specific families, relates metric-locating-dominating sets to other special sets: resolving sets, dominating sets, locating-dominating sets and doubly resolving sets. We first characterize classes of trees according to certain relationships between their metric-location-dominatio…

具有支配参数的图的刻画

Characterizations of the graphs with dominating parameters

本文研究图的支配数、全支配数和孤立数等支配参数，并解答了Hamid等人2016年提出的开放问题。

具有唯一最小支配集的极大二部图

Maximal bipartite graphs with a unique minimum dominating set

本文研究了具有唯一最小支配集的二部图的最大边数界限，提出了紧的边数上界猜想，并在γ=2和n=3γ情况下证明了该界限成立，同时探讨了完美支配与极图构造的关系。

关于具有大最小度的最小韧度图的存在性

On the existence of minimally tough graphs having large minimum degrees

本文构造最小度接近三倍韧度的非正则无爪最小 t-韧度图，推翻平面图及线图情形下的推广 Kriesel 猜想，并补充相关度上界结果。

A Degree Condition for Dominating Cycles in $t$-tough Graphs with $t>1$

Let $G$ be a $t$-tough graph of order $n$ and minimum degree $δ$ with $t>1$. It is proved that if $δ\ge(n-2)/3$ then each longest cycle in $G$ is a dominating cycle.

Perfect dominating sets in the Cartesian products of prime cycles

We study the structure of a minimum dominating set of $C_{2n+1}^n$, the Cartesian product of $n$ copies of the cycle of size $2n+1$, where $2n+1$ is a prime.

Reconfiguration of Minimum Independent Dominating Sets in Graphs

The independent domination number $i(G)$ of a graph $G$ is the minimum cardinality of a maximal independent set of $G$, also called an $i(G)$-set. The $i$-graph of $G$, denoted $\mathcal{I}(G)$, is the graph whose vertices correspond to the $i(G)$-sets, and where two $i(G)$-sets are adjacent if and only if they differ by two adjacent vertices. We show that not all graphs are $i$-graph realizable, that is, given a target graph $H$, there does not necessarily exist a source graph $G$ such that $H$ is isomorphic to $\mathcal{I}(G)$. Examples of such graphs include $K_{4}-e$ and $K_{2,3}$. We buil…

Properties of Partial Dominating Sets of Graphs

A set $S\subseteq V$ is a dominating set of $G$ if every vertex in $V - S$ is adjacent to at least one vertex in $S$. The domination number $γ(G)$ of $G$ equals the minimum cardinality of a dominating set $S$ in $G$; we say that such a set $S$ is a $γ$-set. A generalization of this is partial domination which was introduced in 2017 by Case, Hedetniemi, Laskar, and Lipman [3,2] . In partial domination a set $S$ is a $p$-dominating set if it dominates a proportion $p$ of the vertices in $V$. The p-domination number $γ_{p}(G)$ is the minimum cardinality of a $p$-dominating set in $G$. In this pap…

Minimum Roman Dominating Distance Energy of a Graph

In this correspondence, we introduced the concept of minimum roman dominating distance energy $E_{RDd}(G)$ of a graph $G$ and computed minimum roman dominating distance energy of some standard graphs. Also, we discussed the properties of eigenvalues of a minimum roman dominating distance matrix $A_{RDd}(G).$ Finally, we derived the Upper and lower bounds for $E_{RDd}(G).$

Cubicity of interval graphs and the claw number

Let $G(V,E)$ be a simple, undirected graph where $V$ is the set of vertices and $E$ is the set of edges. A $b$-dimensional cube is a Cartesian product $I_1\times I_2\times...\times I_b$, where each $I_i$ is a closed interval of unit length on the real line. The \emph{cubicity} of $G$, denoted by $\cub(G)$ is the minimum positive integer $b$ such that the vertices in $G$ can be mapped to axis parallel $b$-dimensional cubes in such a way that two vertices are adjacent in $G$ if and only if their assigned cubes intersect. Suppose $S(m)$ denotes a star graph on $m+1$ nodes. We define \emph{claw nu…

关于图的外独立$2$-彩虹控制函数的进一步结果

Further results on outer independent $2$-rainbow dominating functions of graphs

本文证明连通无爪图的OI$2$RD数下界为$n/2$并刻画等号成立图类，解决开放问题；同时给出有根积与冠积的精确公式及笛卡尔积、直接积的上界。

无爪三次图与单位圆盘图中支配数与打包数的改进界

Improved Domination--Packing Bounds in Claw-Free Cubic Graphs and Unit Disk Graphs

研究无爪三次图与单位圆盘图中支配数γ(G)与打包数ρ(G)的比值，将单位圆盘图的界从32改进到16，并证明无桥无爪三次图满足γ(G) ≤ (7/4)ρ(G) + 5/6。

Packing and domination parameters in digraphs

Given a digraph $D=(V,A)$, a set $B\subset V$ is a packing set in $D$ if there are no arcs joining vertices of $B$ and for any two vertices $x,y\in B$ the sets of in-neighbors of $x$ and $y$ are disjoint. The set $S$ is a dominating set (an open dominating set) in $D$ if every vertex not in $S$ (in $V$) has an in-neighbor in $S$. Moreover, a dominating set $S$ is called a total dominating set if the subgraph induced by $S$ has no isolated vertices. The packing sets of maximum cardinality and the (total, open) dominating sets of minimum cardinality in digraphs are studied in this article. We pr…

Adjoining edges to $G\mathbin{\square}H$ to construct a minimal dominating set of size $γ(G)γ(H)$

For graphs $G,H$ it is possible to add $(|V(G)|-γ(G))(|V(H)|-γ(H))$ edges to the Cartesian product $G\mathbin{\square}H$ such that a minimal dominating set $D$ of size $γ(G)γ(H)$ emerges. We hypothesize that $D$ is also a minimum dominating set for the resulting graph and show that this implies Vizing's conjecture.

论图的$nk$-攻击罗马控制数

On the $nk-attack Roman Dominating Number of a Graph

本文研究了图的$k$-攻击罗马控制数，特别是2-攻击罗马控制，求出了特定图类的控制数并讨论了其性质及与其他支配概念的联系。

On a Vizing-type integer domination conjecture

Given a simple graph $G$, a dominating set in $G$ is a set of vertices $S$ such that every vertex not in $S$ has a neighbor in $S$. Denote the domination number, which is the size of any minimum dominating set of $G$, by $γ(G)$. For any integer $k\ge 1$, a function $f : V (G) \rightarrow \{0, 1, . . ., k\}$ is called a \emph{$\{k\}$-dominating function} if the sum of its function values over any closed neighborhood is at least $k$. The weight of a $\{k\}$-dominating function is the sum of its values over all the vertices. The $\{k\}$-domination number of $G$, $γ_{\{k\}}(G)$, is defined to be t…

Maximum matchings and minimum dominating sets in Apollonian networks and extended Tower of Hanoi graphs

The Apollonian networks display the remarkable power-law and small-world properties as observed in most realistic networked systems. Their dual graphs are extended Tower of Hanoi graphs, which are obtained from the Tower of Hanoi graphs by adding a special vertex linked to all its three extreme vertices. In this paper, we study analytically maximum matchings and minimum dominating sets in Apollonian networks and their dual graph- s, both of which have found vast applications in various fields, e.g. structural controllability of complex networks. For both networks, we determine their matching n…

Partitioning an interval graph into subgraphs with small claws

The claw number of a graph $G$ is the largest number $v$ such that $K_{1,v}$ is an induced subgraph of $G$. Interval graphs with claw number at most $v$ are cluster graphs when $v = 1$, and are proper interval graphs when $v = 2$. Let $κ(n,v)$ be the smallest number $k$ such that every interval graph with $n$ vertices admits a vertex partition into $k$ induced subgraphs with claw number at most $v$. Let $\checkκ(w,v)$ be the smallest number $k$ such that every interval graph with claw number $w$ admits a vertex partition into $k$ induced subgraphs with claw number at most $v$. We show that $κ(…

Proper connection number and connected dominating sets

The proper connection number $pc(G)$ of a connected graph $G$ is defined as the minimum number of colors needed to color its edges, so that every pair of distinct vertices of $G$ is connected by at least one path in $G$ such that no two adjacent edges of the path are colored the same, and such a path is called a proper path. In this paper, we show that for every connected graph with diameter 2 and minimum degree at least 2, its proper connection number is 2. Then, we give an upper bound $\frac{3n}{δ+ 1}-1$ for every connected graph of order $n$ and minimum degree $δ$. We also show that for eve…

On the semitotal dominating sets of graphs

A set $D$ of vertices in an isolate-free graph $G$ is a semitotal dominating set of $G$ if $D$ is a dominating set of $G$ and every vertex in $D$ is within distance $2$ from another vertex of $D$.The semitotal domination number of $G$ is the minimum cardinality of a semitotal dominating set of $G$ and is denoted by $γ_{t2}(G)$. In this paper after computation of semitotal domination number of specific graphs, we count the number of this kind of dominating sets of arbitrary size in some graphs.

Domination Value in $P_2 \square P_n$ and $P_2 \square C_n$

A set $D \subseteq V(G)$ is a \emph{dominating set} of a graph $G$ if every vertex of $G$ not in $D$ is adjacent to at least one vertex in $D$. A \emph{minimum dominating set} of $G$, also called a $γ(G)$-set, is a dominating set of $G$ of minimum cardinality. For each vertex $v \in V(G)$, we define the \emph{domination value} of $v$ to be the number of $γ(G)$-sets to which $v$ belongs. In this paper, we find the total number of minimum dominating sets and characterize the domination values for $P_2 \square P_n$ and $P_2 \square C_n$.

Theta 图作为最小独立支配集重配置图的可实现性

The Realizability of Theta Graphs as Reconfiguration Graphs of Minimum Independent Dominating Sets

本文研究 theta 图作为最小独立支配集重配置图（i-图）的可实现性，完整刻画了可实现与不可实现的 theta 图，并进一步刻画了线图、爪形自由图及3-连通三次二部平面图的 i-图性质。

On total dominating sets in graphs

A set $S$ of vertices in a graph $G(V,E)$ is called a dominating set if every vertex $v\in V$ is either an element of $S$ or is adjacent to an element of $S$. A set $S$ of vertices in a graph $G(V,E)$ is called a total dominating set if every vertex $v\in V$ is adjacent to an element of $S$. The domination number of a graph $G$ denoted by $γ(G)$ is the minimum cardinality of a dominating set in $G$. Respectively the total domination number of a graph $G$ denoted by $γ_t(G)$ is the minimum cardinality of a total dominating set in $G$. An upper bound for $γ_t(G)$ which has been achieved by Cocka…

The 3-rainbow index and connected dominating sets

A tree in an edge-colored graph is said to be rainbow if no two edges on the tree share the same color. An edge-coloring of $G$ is called 3-rainbow if for any three vertices in $G$, there exists a rainbow tree connecting them. The 3-rainbow index $rx_3(G)$ of $G$ is defined as the minimum number of colors that are needed in a 3-rainbow coloring of $G$. This concept, introduced by Chartrand et al., can be viewed as a generalization of the rainbow connection. In this paper, we study the 3-rainbow index by using connected three-way dominating sets and 3-dominating sets. We shown that for every co…

Simultaneously Dominating all Spanning Trees of a Graph

We investigate the problem of simultaneously dominating all spanning trees of a given graph. We prove that on 2-connected graphs, a subset of the vertices dominates all spanning trees of the graph if and only if it is a vertex cover. Using this fact we present an exact algorithm that finds a simultaneous dominating set of minimum size using an oracle for finding a minimum vertex cover. The algorithm can be implemented to run in polynomial time on several graph classes, such as bipartite or chordal graphs. We prove that there is no polynomial time algorithm that finds a minimum simultaneous dom…