Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. A $P_t$ is the path on $t$ vertices. A chair is a $P_4$ with an additional vertex adjacent to one of the middle vertices of the $P_4$. A graph $G$ is $k$-vertex-critical if $G$ has chromatic number $k$ but every proper induced subgraph of $G$ has chromatic number less than $k$. In this paper, we prove that there are finitely many 5-vertex-critical $(P_5,chair)$-free graphs.
Seymour's Second Neighborhood Conjecture (SNC) asserts that every oriented graph has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. In this paper, we prove that if $G$ is a graph containing no induced $C_4$, $\overline{C_4}$, $S_3$, chair and $\overline{chair}$, then every oriented graph missing $G$ satisfies this conjecture. As a consequence, we deduce that the conjecture holds for every oriented graph missing a threshold graph, a generalized comb or a star.
Seymour's Second Neighborhood Conjecture asserts that every oriented graph has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. Combs are the graphs having no induced $C_4$, $\overline{C_4}$, $C_5$, chair or $\overline{chair}$. We characterize combs using dependency digraphs. We characterize the graphs having no induced $C_4$, $\overline{C_4}$, chair or $\overline{chair}$ using dependency digraphs. Then we prove that every oriented graph missing a comb satisfies this conjecture. We then deduce that every oriented comb and every oriented threshold graph …
The Maximum Weight Independent Set (MWIS) problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. MWIS is known to be $NP$-complete in general, even under various restrictions. Let $S_{i,j,k}$ be the graph consisting of three induced paths of lengths $i, j, k$ with a common initial vertex. The complexity of the MWIS problem for $S_{1, 2, 2}$-free graphs, and for $S_{1, 1, 3}$-free graphs are open. In this paper, we show that the MWIS problem can solved in polynomial time for ($S_{1, 2, 2}$, $S_{1, 1, 3}$, co-chair)-free graphs, by analyzin…
The exact formula for the average hitting time (HT, as an abbreviation) of simple random walks from one vertex to any other vertex on the square $C^2_N$ of an $N$-vertex cycle graph $C_N$ was given by N. Chair [\textit{Journal of Statistical Physics}, \textbf{154} (2014) 1177-1190]. In that paper, the author gives the expression for the even $N$ case and the expression for the odd $N$ case separately. In this paper, by using an elementary method different from Chair (2014), we give a much simpler single formula for the HT's of simple random walks on $C^2_N$. Our proof is considerably short and…
A natural way of increasing our understanding of NP-complete graph problems is to restrict the input to a special graph class. Classes of $H$-free graphs, that is, graphs that do not contain some graph $H$ as an induced subgraph, have proven to be an ideal testbed for such a complexity study. However, if the forbidden graph $H$ contains a cycle or claw, then these problems often stay NP-complete. A recent complexity study on the $k$-Colouring problem shows that we may still obtain tractable results if we also bound the diameter of the $H$-free input graph. We continue this line of research by …
Let $γ(G)$ denote the domination number of a graph $G$. A vertex $v\in V(G)$ is called a \emph{critical vertex} of $G$ if $γ(G-v)=γ(G)-1$. A graph is called \emph{vertex-critical} if every vertex of it is critical. In this paper, we correspondingly introduce two such definitions: (i) a set $S\subseteq V(G)$ is called a \emph{strong critical vertex-set} of $G$ if $γ(G-S)=γ(G)-|S|$; (ii) a graph $G$ is called \emph{strong $l$-vertex-sets-critical} if $V(G)$ can be partitioned into $l$ strong critical vertex-sets of $G$. Whereafter, we give some properties of strong $l$-vertex-sets-critical graph…
A graph $G$ is said to be $k$-$γ$-vertex critical if the domination numbers $γ(G)$ of $G$ is $k$ and $γ(G - v) < k$ for any vertex $v$ of $G$. Similarly, A graph $G$ is said to be $k$-$γ_{c}$-vertex critical if the connected domination numbers $γ_{c}(G)$ of $G$ is $k$ and $γ_{c}(G - v) < k$ for any vertex $v$ of $G$. The problem of interest is to determine whether or not $2$-connected $k$-$γ$-vertex critical graphs are Hamiltonian. In this paper, for all $k \geq 3$, we provide a $2$-connected $k$-$γ$-vertex critical graph which is non-Hamiltonian. We prove that every $2$-connected $3$-$γ$-vert…
Several games that arise from graph coloring have been introduced and studied. Let $φ$ denote a graph invariant that arises from such a game. If $G$ is a graph and $φ(G-x)\neq φ(G)=k$, $k \geq 1$, holds true for every vertex $x \in V(G)$, then $G$ is called a $k$-$φ$-game-vertex-critical graph. We study the concept of $φ$-game-vertex-criticality for $φ\in \{χ_g, χ_i, χ_{ig}^{A}, χ_{ig}^{AB}\}$, where $χ_g$ denotes the standard game chromatic number, $χ_i$ denotes the indicated game chromatic number and $χ_{ig}^{A}$, $χ_{ig}^{AB}$ denote two versions of the independence game chromatic number. S…
A graph of order $n$ is $p$-factor-critical, where $p$ is an integer of the same parity as $n$, if the removal of any set of $p$ vertices results in a graph with a perfect matching. 1-Factor-critical graphs and 2-factor-critical graphs are factor-critical graphs and bicritical graphs, respectively. It is well known that every connected vertex-transitive graph of odd order is factor-critical and every connected non-bipartite vertex-transitive graph of even order is bicritical. In this paper, we show that a simple connected vertex-transitive graph of odd order at least 5 is 3-factor-critical if …
A vertex subset $S$ of a graph $G$ is a dominating set if every vertex of $G$ either belongs to $S$ or is adjacent to a vertex of $S$. The cardinality of a smallest dominating set is called the dominating number of $G$ and is denoted by $γ(G)$. A graph $G$ is said to be $γ$- vertex-critical if $γ(G-v)< γ(G)$, for every vertex $v$ in $G$. Let $G$ be a 2-connected $K_{1,5}$-free 3-vertex-critical graph. For any vertex $v \in V(G)$, we show that $G-v$ has a perfect matching (except two graphs), which is a conjecture posed by Ananchuen and Plummer.
A graph of order $n$ is $p$-factor-critical, where $p$ is an integer of the same parity as $n$, if the removal of any set of $p$ vertices results in a graph with a perfect matching. 1-factor-critical graphs and 2-factor-critical graphs are well-known factor-critical graphs and bicritical graphs, respectively. It is known that if a connected vertex-transitive graph has odd order, then it is factor-critical, otherwise it is elementary bipartite or bicritical. In this paper, we show that a connected vertex-transitive non-bipartite graph of even order at least 6 is 4-factor-critical if and only if…
Given a graph $G$, a function $c:V(G)\longrightarrow \{1,\ldots,k\}$ with the property that $c(u)=c(v)=i$ implies that the distance between $u$ and $v$ is greater than $i$, is called a $k$-packing coloring of $G$. The smallest integer $k$ for which there exists a $k$-packing coloring of $G$ is called the packing chromatic number of $G$, and is denoted by $χ_ρ$. Packing chromatic vertex-critical graphs are the graphs $G$ for which $χ_ρ(G-x) < χ_ρ(G)$ holds for every vertex $x$ of $G$. A graph $G$ is called a packing chromatic critical graph if for every proper subgraph $H$ of $G$, $χ_ρ(H) < χ_ρ…
We prove that $K_{χ(G)}$ is the only critical graph $G$ with $χ(G) \geq Δ(G) \geq 6$ and $ω(\mathcal{H}(G)) \leq \left \lfloor \frac{Δ(G)}{2} \right \rfloor - 2$. Here $\mathcal{H}(G)$ is the subgraph of $G$ induced on the vertices of degree at least $χ(G)$. Setting $ω(\mathcal{H}(G)) = 1$ proves a conjecture of Kierstead and Kostochka.
Let $γ(G)$ be the domination number of a graph $G$. A graph $G$ is \emph{domination-vertex-critical}, or \emph{$γ$-vertex-critical}, if $γ(G-v)< γ(G)$ for every vertex $v \in V(G)$. In this paper, we show that: Let $G$ be a $γ$-vertex-critical graph and $γ(G)=3$. (1) If $G$ is of even order and $K_{1,6}$-free, then $G$ has a perfect matching; (2) If $G$ is of odd order and $K_{1,7}$-free, then $G$ has a near perfect matching with only three exceptions. All these results improve the known results.
A set $D$ of vertices of a graph $G$ is a dominating set if each vertex of $V(G)\setminus D$ is adjacent to some vertex of $D$. The domination number of $G$, $γ(G)$, is the minimum cardinality of a dominating set of $G$. A graph $G$ is called domination vertex critical, or just $γ$-critical if removal of any vertex decreases the domination number. A graph $G$ is called domination vertex stable, or just $γ$-stable, if removal of any vertex does not decrease the domination number. For an even integer $n\ge 2$ and $1\le Δ\le \lfloor \log_2n \rfloor$, a Knödel graph $W_{Δ,n}$ is a $Δ$-regular bipa…
A note on vertex-critical induced subgraphs of shift graphs
University of West Bohemia（西波希米亚大学）；CNRS（法国国家科学研究中心）；IRIF（IRIF实验室）；Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学物理学院）；Faculty of Information Studies in Novo Mesto（诺沃梅斯托信息研究学院）；Rudolfovo – Science and Technology Centre（鲁多维福科学与技术中心）；University of Southern Denmark（南丹麦大学）
Let $G$ be a simple graph with maximum degree $Δ$. A classic result of Vizing shows that $χ'(G)$, the chromatic index of $G$, is either $Δ$ or $Δ+1$. We say $G$ is of \emph{Class 1} if $χ'(G)=Δ$, and is of \emph{Class 2} otherwise. A graph $G$ is \emph{$Δ$-critical} if $χ'(G)=Δ+1$ and $χ'(H)<Δ+1$ for every proper subgraph $H$ of $G$, and is \emph{overfull} if $|E(G)|>Δ\lfloor (|V(G)|-1)/2 \rfloor$. Clearly, overfull graphs are Class 2. Hilton and Zhao in 1997 conjectured that if $G$ is obtained from an $n$-vertex $Δ$-regular Class 1 graph with maximum degree greater than $n/3$ by splitting a v…
We show that every k-dichromatic vertex-critical digraph on at most 2k-2 vertices has a disconnected complement. This answers a question of Bang-Jensen et al., and generalises a classical theorem of Gallai on undirected vertex-critical graphs.
The vertex v of a graph G is called a 1-critical-vertex for the maximum genus of the graph, or for simplicity called 1-critical-vertex, if G-v is a connected graph and °M(G - v) = °M(G) - 1. In this paper, through the joint-tree model, we obtained some types of 1-critical-vertex, and get the upper embeddability of the Spiral Snm .
In this paper, we consider various types of domination vertex critical graphs, including total domination vertex critical graphs, independent domination vertex critical graphs and connected domination vertex critical graphs. We provide upper bounds on the diameter of them, two of which are sharp.
University of West Bohemia（西波希米亚大学）；University of Maribor（马里博尔大学）；Institute of Mathematics, Physics and Mechanics（数学、物理与力学研究所）；University of Ljubljana（卢布尔雅那大学）
A $k$-$γ_{c}$-edge critical graph is a graph $G$ with the connected domination number $γ_{c}(G) = k$ and $γ_{c}(G + uv) < k$ for every $uv \in E(\overline{G})$. Further, a $2$-connected graph $G$ is said to be $k$-$γ_{c}$-vertex critical if $γ_{c}(G) = k$ and $γ_{c}(G - v) < k$ for all $v \in V(G)$. A maximal $k$-$γ_{c}$-vertex critical graph is a graph which are both $k$-$γ_{c}$-edge critical and $k$-$γ_{c}$-vertex critical. Let $κ, δ, ω$ and $α$ be respectively connectivity minimum degree, clique number and independence number. In this paper, we prove that every maximal $3$-$γ_{c}$-vertex cr…
Let $γ_t(G)$ be the total domination number of graph $G$, a graph $G$ is $k$-total domination vertex critical (or\ just\ $k$-$γ_t$-critical) if $γ_t(G)=k$, and for any vertex $v$ of $G$ that is not adjacent to a vertex of degree one, $γ_t(G-v)=k-1$. Mojdeh and Rad \cite{MR06} proposed an open problem: Does there exist a 3-$γ_t$-critical graph $G$ of order $Δ(G)+3$ with $Δ(G)$ odd? In this paper, we prove that there exists a 3-$γ_t$-critical graph $G$ of order $Δ(G)+3$ with odd $Δ(G)\geq 9$.
A graph $G$ is $k$-vertex-critical if $χ(G)=k$ but $χ(G-v)<k$ for all $v\in V(G)$ where $χ(G)$ denotes the chromatic number of $G$. We show that there are only finitely many $k$-critical $(P_3+\ell P_1)$-free graphs for all $k$ and all $\ell$. Together with previous results, the only graphs $H$ for which it is unknown if there are an infinite number of $k$-vertex-critical $H$-free graphs is $H=(P_4+\ell P_1)$ for all $\ell\ge 1$. We consider a restriction on the smallest open case, and show that there are only finitely many $k$-vertex-critical (gem, co-gem)-free graphs for all $k$, where gem$=…
The existence problem of the total domination vertex critical graphs has been studied in a series of articles. The aim of the present article is twofold. First, we settle the existence problem with respect to the parities of the total domination number m and the maximum degree Delta : for even m except m=4, there is no m-gamma_t-critical graph regardless of the parity of Delta; for m=4 or odd m \ge 3 and for even Delta, an m-gamma_t-critical graph exists if and only if Delta \ge 2 \lfloor \frac{m-1}{2}\rfloor; for m=4 or odd m \ge 3 and for odd Delta, if Delta \ge 2\lfloor \frac{m-1}{2}\rfloor…
Define a boundary point of a graph which is embedded in the Euclidean plane a vertex which is incident to only one edge. In this paper we consider graphs which are embedded in the Euclidean plane with a finite number of boundary points. The simple geometric condition we impose on them is that the sum of unit vectors of edges extending from each non-boundary vertex will be equal to zero. We call such a graph a critical graph and ask to maximise the number of vertices of critical graphs with a given size of boundary. The main results of this paper give a sharp upper bound for the maximum number …
Schrijver graphs are vertex-color-critical subgraphs of Kneser graphs having the same chromatic number. They also share the value of their fractional chromatic number but Schrijver graphs are not critical for that. Here we present an induced subgraph of every Schrijver graph that is vertex-critical with respect to the fractional chromatic number. These subgraphs turn out to be isomorphic with certain circular complete graphs. We also characterize the critical edges within this subgraph.
A graph G is equimatchable if every maximal matching of G has the same cardinality. In this paper, we investigate equimatchable graphs such that the removal of any edge harms the equimatchability, called edge-critical equimatchable graphs (ECE-graphs). We show that apart from two simple cases, namely bipartite ECE-graphs and even cliques, all ECE-graphs are 2-connected factor-critical. Accordingly, we give a characterization of factor-critical ECE-graphs with connectivity 2. Our result provides a partial answer to an open question posed by Levit and Mandrescu on the characterization of well-co…
A $p$-improper interval graph is an interval graph that has an interval representation in which no interval contains more than $p$ other intervals. A critical $p$-improper interval graph is $p-1$ improper when any vertex is removed. In this paper we investigate the spectrum of impropriety of critical $p$-improper interval graphs upon the removal of a single vertex, which is informally known as the stability of the graph.
Let $G_1\sq G_2$ denote the graph obtained from $G_1+G_2$ by adding new edges from each vertex of $G_1$ to every vertex of $G_2$. In this paper, the critical groups of the graphs $K_m\sq P_n$$(n\geq4)$ and $P_m\sq P_n$$(m\geq4,n\geq5)$ are determined.
A vertex subset $S$ of a graph $G$ is a double dominating set of $G$ if $|N[v]\cap S|\geq 2$ for each vertex $v$ of $G$, where $N[v]$ is the set of the vertex $v$ and vertices adjacent to $v$. The double domination number of $G$, denoted by $γ_{\times 2}(G)$, is the cardinality of a smallest double dominating set of $G$. A graph $G$ is said to be double domination edge critical if $γ_{\times 2}(G+e)<γ_{\times 2}(G)$ for any edge $e \notin E$. A double domination edge critical graph $G$ with $γ_{\times 2}(G)=k$ is called $k$-$γ_{\times 2}(G)$-critical. A graph $G$ is $r$-factor-critical if $G-S…
Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ be the path on $t$ vertices. A graph $G$ is $k$-vertex-critical if $G$ has chromatic number $k$ but every proper induced subgraph of $G$ has chromatic number less than $k$. The study of $k$-vertex-critical graphs for graph classes is an important topic in algorithmic graph theory because if the number of such graphs that are in a given hereditary graph class is finite, then there is a polynomial-time algorithm to decide if a graph in the class is $(k-1)$-…
A Roman domination function on a graph G is a function $r:V(G)\to \{0,1,2\}$ satisfying the condition that every vertex $u$ for which $r(u)=0$ is adjacent to at least one vertex $v$ for which $r(v)=2$. The weight of a Roman function is the value $r(V(G))=\sum_{u\in V(G)}r(u)$. The Roman domination number $γ_R(G)$ of $G$ is the minimum weight of a Roman domination function on $G$. "Roman Criticality" has been defined in general as the study of graphs where the Roman domination number decreases when removing an edge or a vertex of the graph. In this paper we give further results in this topic as…
The tree-depth of $G$ is the smallest value of $k$ for which a labeling of the vertices of $G$ with elements from $\{1,\dots,k\}$ exists such that any path joining two vertices with the same label contains a vertex having a higher label. The graph $G$ is $k$-critical if it has tree-depth $k$ and every proper minor of $G$ has smaller tree-depth. Motivated by a conjecture on the maximum degree of $k$-critical graphs, we consider the property of 1-uniqueness, wherein any vertex of a critical graph can be the unique vertex receiving label 1 in an optimal labeling. Contrary to an earlier conjecture…
A k-ranking of a graph G is a labeling of the vertices of G with values from {1,...,k} such that any path joining two vertices with the same label contains a vertex having a higher label. The tree-depth of G is the smallest value of k for which a k-ranking of G exists. The graph G is k-critical if it has tree-depth k and any proper minor of G has smaller tree-depth, and it is 1-unique if for every vertex v in G, there exists an optimal ranking of G in which v is the unique vertex with label 1. We present several classes of graphs that are both k-critical and 1-unique, providing examples that s…
Let $G$ be a connected simple graph of order $n$ and let $Δ(G)$ and $χ'(G)$ denote the maximum degree and chromatic index of $G$, respectively. Vizing proved that $χ'(G)=Δ(G)$ or $Δ(G)+1$. Following this result, $G$ is called $Δ$-critical if $χ'(G)=Δ(G)+1$ and $χ'(G-e)=Δ(G)$ for every $e\in E(G)$. In 1968, Vizing conjectured that if $G$ is an $n$-vertex $Δ$-critical graph, then the independence number $α(G)\le n/2$. Furthermore, he conjectured that, in fact, $G$ has a 2-factor. Luo and Zhao showed that if $G$ is an $n$-vertex $Δ$-critical graph with $Δ(G)\ge n/2$, then $α(G)\le n/2$. More rece…
The Turán number of a graph $H$, $\text{ex}(n,H)$, is the maximum number of edges in an $n$-vertex graph that does not contain $H$ as a subgraph. For a vertex $v$ and a multi-set $\mathcal{F}$ of graphs, the suspension $\mathcal{F}+v$ of $\mathcal{F}$ is the graph obtained by connecting the vertex $v$ to all vertices of $F$ for each $F\in \mathcal{F}$. For two integers $k\ge1$ and $r\ge2$, let $H_i$ be a graph containing a critical edge with chromatic number $r$ for any $i\in\{1,\ldots,k\}$, and let $H=\{H_1,\ldots,H_k\}+v$. In this paper, we determine $\text{ex}(n, H)$ and characterize all th…
For $k \geq 3$, we prove (i) there is a finite number of $k$-vertex-critical $(P_2+\ell P_1)$-free graphs and (ii) $k$-vertex-critical $(P_3+P_1)$-free graphs have at most $2k-1$ vertices. Together with previous research, these results imply the following characterization where $H$ is a graph of order four: There is a finite number of $k$-vertex-critical $H$-free graphs for fixed $k \geq 5$ if and only if $H$ is one of $\overline{K_4}, P_4, P_2 + 2P_1$, or $P_3 + P_1$. Our results imply the existence of new polynomial-time certifying algorithms for deciding the $k$-colorability of $(P_2+\ell P…
In the total domination game played on a graph $G$, players Dominator and Staller alternately select vertices of $G$, as long as possible, such that each vertex chosen increases the number of vertices totally dominated. Dominator (Staller) wishes to minimize (maximize) the number of vertices selected. The game total domination number, $γ_{\rm tg}(G)$, of $G$ is the number of vertices chosen when Dominator starts the game and both players play optimally. If a vertex $v$ of $G$ is declared to be already totally dominated, then we denote this graph by $G|v$. In this paper the total domination gam…
Every fork-free graph is perfectly weight divisible
School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；School of Mathematics and Statistics, Jiangsu Normal University（江苏师范大学数学与统计学院）；Department of Mathematics, Hong Kong University of Science and Technology（香港科技大学数学系）
The domination number of a graph $G$, denoted by $γ(G)$, is the minimum cardinality of a dominating set of $G$. A vertex of a graph is called critical if its deletion decreases the domination number, and a graph is called critical if its all vertices are critical. A graph $G$ is called weak bicritical if for every non-critical vertex $x\in V(G)$, $G-x$ is a critical graph with $γ(G-x)=γ(G)$. In this paper, we characterize the connected weak bicritical graphs $G$ whose diameter is exactly $2γ(G)-2$. This is a generalization of some known results concerning the diameter of graphs with a dominati…
Ege University（伊兹密尔爱琴海大学）；University of Ljubljana（卢布尔雅那大学）；Institute of Mathematics, Physics and Mechanics（数学、物理与力学研究所）；Gebze Technical University（格布泽理工大学）
