一个图$G$中没有孤立点的顶点序列称为全控制序列，如果序列中的每个顶点都完全控制至少一个不完全被序列中的前一个顶点控制的顶点，并且在末端$G$的所有顶点都是全控制的（根据定义，一个顶点完全控制它的邻居）。全控制序列的最大长度称为$G$的Grundy全控制数，$γ_{\rm gr}^t(G)$，如[B. Brešar，M.A. Henning和D. F. Rall，Total dominant sequences in graphs，Discrete Math.339（2016），1165- 1676].在本文中，我们继续调查这一概念，主要是从算法的角度来看。虽然它是已知的，决策版本的问题是NP-完全的二分图，我们表明，这也是真的，如果我们限制分裂图。基于公式$γ_{\rm gr}^t(T)=2τ(T)$，给出了一个确定任意树$T$的Grundy全控制数的线性时间算法，其中$τ(T)$是$T$的顶点覆盖数.对二部距离遗传图给出了一个类似的有效算法。使用图的模块化分解，我们提出了一个框架，获得多项式算法，这个问题的类图具有相对简单的模块化子图。特别地，给出了一个确定$P_4$ -整洁图的Grundy全控制数的线性算法.此外，我们证明了一个实现的结果，展示了一个家庭的图$G_k$，使$γ_{\rm gr}^t(G_k)=k$，对任何$k\in{\mathbb{Z}^+}\setminus\{1,3\}$，并表明，有没有图$G$与$γ_{\rm gr}^t(G)\in \{1,3\}$。我们还提出了这样一个家庭，它有最小可能的顺序和大小之间的所有图的Grundy全控制数等于$k$。
英文摘要：
A sequence of vertices in a graph $G$ with no isolated vertices is called a total dominating sequence if every vertex in the sequence totally dominates at least one vertex that was not totally dominated by preceding vertices in the sequence, and, at the end all vertices of $G$ are totally dominated (by definition a vertex totally dominates its neighbors). The maximum length of a total dominating sequence is called the Grundy total domination number, $γ_{\rm gr}^t(G)$, of $G$, as introduced in [B. Brešar, M.A. Henning, and D. F. Rall, Total dominating sequences in graphs, Discrete Math. 339 (2016), 1165--1676]. In this paper we continue the investigation of this concept, mainly from the algorithmic point of view. While it was known that the decision version of the problem is NP-complete in bipartite graphs, we show that this is also true if we restrict to split graphs. A linear time algorithm for determining the Grundy total domination number of an arbitrary tree $T$ is presented, based on the formula $γ_{\rm gr}^t(T)=2τ(T)$, where $τ(T)$ is the vertex cover number of $T$. A similar efficient algorithm is presented for bipartite distance-hereditary graphs. Using the modular decomposition of a graph, we present a frame for obtaining polynomial algorithms for this problem in classes of graphs having relatively simple modular subgraphs. In particular, a linear algorithm for determining the Grundy total domination number of $P_4$-tidy graphs is presented. In addition, we prove a realization result by exhibiting a family of graphs $G_k$ such that $γ_{\rm gr}^t(G_k)=k$, for any $k\in{\mathbb{Z}^+}\setminus\{1,3\}$, and showing that there are no graphs $G$ with $γ_{\rm gr}^t(G)\in \{1,3\}$. We also present such a family, which has minimum possible order and size among all graphs with Grundy total domination number equal to $k$.
Let $d,n$ be positive integers such that $d<n$, and let $X^d_n$ be a Cayley graph generated by a transposition tree of diameter $d$. It is known that every $X^d_n$ with $d<3$ splits into efficient dominating sets. The main result of this paper is that $X^3_n$ does not have efficient dominating sets.
In [M. M. Kanté, V. Limouzy, A. Mary, and L. Nourine. On the enumeration of minimal dominating sets and related notions. SIAM Journal on Discrete Mathematics, 28(4):1916-1929, 2014] the authors give an $O(n+m)$ delay algorithm based on neighborhood inclusions for the enumeration of minimal dominating sets in split and $P_6$-free chordal graphs. In this paper, we investigate generalizations of this technique to $P_k$-free chordal graphs for larger integers $k$. In particular, we give $O(n+m)$ and $O(n^3\cdot m)$ delays algorithms in the classes of $P_7$-free and $P_8$-free chordal graphs. As fo…
Many well-known NP-hard algorithmic problems on directed graphs resist efficient parametrisations with most known width measures for directed graphs, such as directed treewidth, DAG-width, Kelly-width and many others. While these focus on measuring how close a digraph is to an oriented tree resp. a directed acyclic graph, in this paper, we investigate directed modular width as a parameter, which is closer to the concept of clique-width. We investigate applications of modular decompositions of directed graphs to a wide range of algorithmic problems and derive FPT-algorithms for several well-kno…
On $\{k\}$-Roman graphs: complexity of recognition and the case of split graphs
Faculty of Information Studies in Novo Mesto, Slovenia（新梅斯托信息学院）；FAMNIT, University of Primorska, Koper, Slovenia（普里莫尔大学 FAMNIT 学院）；IAM, University of Primorska, Koper, Slovenia（普里莫尔大学 IAM 研究所）；Universidad Nacional de Rosario, Argentina（罗萨里奥国立大学）；CONICET, Argentina（阿根廷国家科学研究和技术委员会）
Let $G = (V,E)$ be a simple, undirected and connected graph. A connected (total) dominating set $S \subseteq V$ is a secure connected (total) dominating set of $G$, if for each $ u \in V \setminus S$, there exists $v \in S$ such that $uv \in E$ and $(S \setminus \lbrace v \rbrace) \cup \lbrace u \rbrace $ is a connected (total) dominating set of $G$. The minimum cardinality of a secure connected (total) dominating set of $G$ denoted by $ γ_{sc} (G) (γ_{st}(G))$, is called the secure connected (total) domination number of $G$. In this paper, we show that the decision problems corresponding to s…
For a simple graph $G=(V,E)$ without any isolated vertex, a cosecure dominating set $D$ of $G$ satisfies the following two properties (i) $S$ is a dominating set of $G$, (ii) for every vertex $v \in S$ there exists a vertex $u \in V \setminus S$ such that $uv \in E$ and $(S \setminus \{v\}) \cup \{u\}$ is a dominating set of $G$. The minimum cardinality of a cosecure dominating set of $G$ is called cosecure domination number of $G$ and is denoted by $γ_{cs}(G)$. The Minimum Cosecure Domination problem is to find a cosecure dominating set of a graph $G$ of cardinality $γ_{cs}(G)$. The decision …
While there has been some progress on the decomposition of Kronecker products of characters of the symmetric groups in recent times, results on the symmetric and alternating part of Kronecker squares are still scarce. Here, new results (and conjectures) are presented on this splitting of the squares that contribute to a refined understanding of the Kronecker squares. Furthermore, connections to 2-modular decomposition numbers, Catalan combinatorics, and to the Saxl conjecture are discussed which further motivate the study of these splittings.
