A set $R \subseteq V(G)$ is a resolving set of a graph $G$ if for all distinct vertices $v,u \in V(G)$ there exists an element $r \in R$ such that $d(r,v) \neq d(r,u)$. The metric dimension $\dim(G)$ of the graph $G$ is the minimum cardinality of a resolving set of $G$. A resolving set with cardinality $\dim(G)$ is called a metric basis of $G$. We consider vertices that are in all metric bases, and we call them basis forced vertices. We give several structural properties of sparse and dense graphs where basis forced vertices are present. In particular, we give bounds for the maximum number of edges in a graph containing basis forced vertices. Our bound is optimal whenever the number of basis forced vertices is even. Moreover, we provide a method of constructing fairly sparse graphs with basis forced vertices. We also study vertices which are in no metric basis in connection to cut-vertices and pendants. Furthermore, we show that deciding whether a vertex is in all metric bases is co-NP-hard, and deciding whether a vertex is in no metric basis is NP-hard.
In a graph G, a dissociation set is a subset of vertices which induces a subgraph with vertex degree at most 1. Finding a dissociation set of maximum cardinality in a graph is NP-hard even for bipartite graphs and is called the maximum dissociation set problem. The complexity of maximum dissociation set problem in various subclasses of graphs has been extensively studied in the literature. In this paper, we study the maximum dissociation problem from different perspectives and characterize the vertices belonging to all maximum dissociation sets, and to no maximum dissociation set of a tree. We…
We say that a graph $G$ is $(2,m)$-linked if, for any distinct vertices $a_1,\ldots, a_m, b_1,b_2$ in $G$, there exist vertex disjoint connected subgraphs $A,B$ of $G$ such that $\{a_1, \ldots, a_m\}$ is contained in $A$ and $\{b_1,b_2\}$ is contained in $B$. A fundamental result in structural graph theory is the characterization of $(2,2)$-linked graphs, with different versions obtained independently by Robertson and Chakravarty, Seymour, and Thomassen. It appears to be very difficult to characterize $(2,m)$-linked graphs for $m\ge 3$. In this paper, we provide a partial characterization of $…
A set of vertices $S\subseteq V(G)$ is a basis or resolving set of a graph $G$ if for each $x,y\in V(G)$ there is a vertex $u\in S$ such that $d(x,u)\neq d(y,u)$. A basis $S$ is a fault-tolerant basis if $S\setminus \{x\}$ is a basis for every $x \in S$. The fault-tolerant metric dimension (FTMD) $β'(G)$ of $G$ is the minimum cardinality of a fault-tolerant basis. It is shown that each twin vertex of $G$ belongs to every fault-tolerant basis of $G$. As a consequence, $β'(G) = n(G)$ iff each vertex of $G$ is a twin vertex, which corrects a wrong characterization of graphs $G$ with $β'(G) = n(G)…
A walk $W$ between vertices $u$ and $v$ of a graph $G$ is called a {\em tolled walk between $u$ and $v$} if $u$, as well as $v$, has exactly one neighbour in $W$. A set $S \subseteq V(G)$ is {\em toll convex} if the vertices contained in any tolled walk between two vertices of $S$ are contained in $S$. The {\em toll convex hull of $S$} is the minimum toll convex set containing~$S$. The {\em toll hull number of $G$} is the minimum cardinality of a set $S$ such that the toll convex hull of $S$ is $V(G)$. The main contribution of this work is an algorithm for computing the toll hull number of a g…
On the Maximum Number of Vertices that Belong to Every Metric Basis
Turku Collegium for Science, Medicine and Technology (TCSMT), University of Turku（图尔库科学技术学院）；Department of Mathematics and Statistics, University of Turku（图尔库大学数学与统计系）
The domination game is played on a graph $G$ by two players, Dominator and Staller, who alternate in selecting vertices until each vertex in the graph $G$ is contained in the closed neighbourhood of the set of selected vertices. Dominator's aim is to reach this state in as few moves as possible, whereas Staller wants the game to last as long as possible. In this paper, we prove that if $G$ has $n$ vertices and minimum degree at least 2, then Dominator has a strategy to finish the domination game on $G$ within $10n/17+1/17$ moves, thus making progress towards a conjecture by Bujt{á}s, Iršič and…
Let $G$ be a connected graph and $d(a,b)$ be the distance between the vertices $a$ and $b$. A subset $U =\{u_1,u_2,\cdots,u_k\}$ of the vertices is called a resolving set for $G$ if for every two distinct vertices $a,b \in V(G)$, there is a vertex $u_ξ\in U$ such that $d(a,u_ξ)\neq d(b,u_ξ)$. A resolving set containing a minimum number of vertices is called a metric basis for $G$ and the number of vertices in a metric basis is its metric dimension denoted by $dim(G)$. A resolving set $U$ for $G$ is fault-tolerant if $U \setminus \{u\}$ is also a resolving set, for each $u \in U$, and the fault…
Let $R$ be a commutative ring with identity. The co-maximal ideal graph of $R$, denoted by $Γ(R)$, is a simple graph whose vertices are proper ideals of $R$ which are not contained in the Jacobson radical of $R$ and two distinct vertices $I, J$ are adjacent if and only if $I+J=R$. In this paper, we use Gallai$^{^,}$s Theorem and the concept of strong resolving graph to compute the strong metric dimension for co-maximal ideal graphs of commutative rings. Explicit formulae for the strong metric dimension, depending on whether the ring is reduced or not, are established.
We prove that for a large family of product graphs, and for Kneser graphs $K(n,αn)$ with fixed $α<1/2$, the following holds. Any set of vertices that spans a small proportion of the edges in the graph can be made independent by removing a small proportion of the vertices of the graph. This allows us to strengthen the results of [DinurFR06] and [DinurF09], and show that any independent set in these graphs is almost contained in an independent set which depends on few coordinates. Our proof is inspired by, and follows some of the main ideas of, Fox's proof of the graph removal lemma [Fox11].
A non-zero component graph $G(\mathbb{V})$ associated to a finite vector space $\mathbb{V}$ is a graph whose vertices are non-zero vectors of $\mathbb{V}$ and two vertices are adjacent, if their corresponding vectors have at least one non-zero component common in their linear combination of basis vectors. In this paper, we extend the study of properties of automorphisms of non-zero component graphs. We prove that every permutation of basis vectors can be extended to an automorphism of $G(\mathbb{V})$. We prove that the symmetric group of basis vectors of $\mathbb{V}$ is isomorphic to the autom…
For a graph G, a hypergraph H is called Berge-G if there is a hypergraph H', isomorphic to H, containing all vertices of G, so that e is contained in f(e) for each edge e of G, where f is a bijection between E(G) and E(H'). The set of all Berge-G hypergraphs is denoted B(G). A hypergraph H is called Berge-G saturated if it does not contain any subhypergraph from B(G), but adding any new hyperedge of size at least 2 to H creates such a subhypergraph. Each Berge-G saturated hypergraph has at least |E(G)|-1 hyperedges. We show that for each graph G that is not a certain star and for any n at leas…
Let $G$ be a finite connected simple graph with $n$ vertices and $m$ edges. We show that, when $G$ is not bipartite, the number of $4$-cycles contained in $G$ is at most $\binom{m-n+1}{2}$. We further provide a short combinatorial proof of the bound $\binom{m-n+2}{2}$ which holds for bipartite graphs.
An $\mathcal{F}$-saturated $r$-graph is a maximal $r$-graph not containing any member of $\mathcal{F}$ as a subgraph. Let $\mathcal{K}_{\ell + 1}^{r}$ be the collection of all $r$-graphs $F$ with at most $\binom{\ell+1}{2}$ edges such that for some $\left(\ell+1\right)$-set $S$ every pair $\{u, v\} \subset S$ is covered by an edge in $F$. Our first result shows that for each $\ell \geq r \geq 2$ every $\mathcal{K}_{\ell+1}^{r}$-saturated $r$-graph on $n$ vertices with $t_{r}(n, \ell) - o(n^{r-1+1/\ell})$ edges contains a complete $\ell$-partite subgraph on $(1-o(1))n$ vertices, which extends a…
A graph $G$ is said to be $k$-$γ_{c}$-critical if the connected domination number $γ_{c}(G)$ is equal to $k$ and $γ_{c}(G + uv) < k$ for any pair of non-adjacent vertices $u$ and $v$ of $G$. Let $ζ$ be the number of cut vertices of $G$ and let $ζ_{0}$ be the maximum number of cut vertices that can be contained in one block. For an integer $\ell \geq 0$, a graph $G$ is $\ell$-factor critical if $G - S$ has a perfect matching for any subset $S$ of vertices of size $\ell$. It was proved that, for $k \geq 3$, every $k$-$γ_{c}$-critical graph has at most $k - 2$ cut vertices and the graphs with max…
For a hypergraph $\mathcal{H}$, define the minimum positive codegree $δ_i^+(\mathcal{H})$ to be the largest integer $k$ such that every $i$-set which is contained in at least one edge of $\mathcal{H}$ is contained in at least $k$ edges. For $1\le s\le k,t$ and $t\le r$, we prove that for $n$-vertex $t$-intersecting $r$-graphs $\mathcal{H}$ with $δ_{r-s}^+(\mathcal{H})>{k-1\choose s}$, the unique hypergraph with the maximum number of edges is the hypergraph $\mathcal{H}$ consisting of every edge which intersects a set of size $2k-2s+t$ in at least $k-s+t$ vertices provided $n$ is sufficiently l…
By the theorem of Mantel $[5]$ it is known that a graph with $n$ vertices and $\lfloor \frac{n^{2}}{4} \rfloor+1$ edges must contain a triangle. A theorem of Erdős gives a strengthening: there are not only one, but at least $\lfloor\frac{n}{2}\rfloor$ triangles. We give a further improvement: if there is no vertex contained by all triangles then there are at least $n-2$ of them. There are some natural generalizations when $(a)$ complete graphs are considered (rather than triangles), $(b)$ the graph has $t$ extra edges (not only one) or $(c)$ it is supposed that there are no $s$ vertices such t…
Let $n$, $k$ and $l$ be integers with $1\leq k<l\leq n-1$. The set-inclusion graph $G(n,k,l)$ is the graph whose vertex set consists of all $k$- and $l$-subsets of $[n]=\{1,2,\ldots,n\}$, where two distinct vertices are adjacent if one of them is contained in another. In this paper, we determine the spectrum and automorphism group of $G(n,k,l)$, respectively.
Let $G$ be a simple and connected graph with vertex set $V(G)$. A vertex $w\in V(G)$ strongly resolves two vertices $u,v \in V(G)$ if $v$ belongs to a shortest $u-w$ path or $u$ belongs to a shortest $v-w$ path. A set $W \subseteq V(G)$ is a strong resolving set for $G$ if every pair of vertices of $G$ is strongly resolved by some vertex of $W$. A strong metric basis of $G$ is a strong resolving set for $G$ with minimum cardinality. The strong metric dimension of $G$, denoted by $sdim(G)$, is the cardinality of a strong metric basis of $G$. In this paper we compute the strong metric dimension …
一个超图H的独立数是指不包含H的边的最大顶点集的大小。本文证明了n-顶点（r+1）-一致超图的独立数的新的精确界，其中每个r-元集至多包含在d条边中，其中0 < d < n/（logn）^{3r^2}.我们的相对较短的证明扩展了Shearer的方法。我们给出了一个应用程序的超图Ramsey数涉及独立的邻域。
In an edge-colored graph $(G,c)$, let $d^c(v)$ denote the number of colors on the edges incident with a vertex $v$ of $G$ and $δ^c(G)$ denote the minimum value of $d^c(v)$ over all vertices $v\in V(G)$. A cycle of $(G,c)$ is called proper if any two adjacent edges of the cycle have distinct colors. An edge-colored graph $(G,c)$ on $n\geq 3$ vertices is called properly vertex-pancyclic if each vertex of $(G,c)$ is contained in a proper cycle of length $\ell$ for every $\ell$ with $3 \le \ell \le n$. Fujita and Magnant conjectured that every edge-colored complete graph on $n\geq 3$ vertices with…
On the number of edges of restricted matchstick graphs
Eötvös University（厄特沃什大学）；Alfréd Rényi Institute of Mathematics（阿尔弗雷德·雷尼数学研究所）；Department of Mathematics, London School of Economics and Political Science（伦敦政治经济学院数学系）；Budapest University of Technology and Economics（布达佩斯技术与经济大学）
Considering a natural generalization of the Ruzsa-Szemerédi problem, we prove that for any fixed positive integers $r,s$ with $r<s$, there are graphs on $n$ vertices containing $n^{r}e^{-O(\sqrt{\log{n}})}=n^{r-o(1)}$ copies of $K_s$ such that any $K_r$ is contained in at most one $K_s$. We also give bounds for the generalized rainbow Turán problem $\operatorname{ex}(n, H,$rainbow-$F)$ when $F$ is complete. In particular, we answer a question of Gerbner, Mészáros, Methuku and Palmer, showing that there are properly edge-coloured graphs on $n$ vertices with $n^{r-1-o(1)}$ copies of $K_r$ such t…
