Given an ordered partition $Π=\{P_1,P_2, ...,P_t\}$ of the vertex set $V$ of a connected graph $G=(V,E)$, the \emph{partition representation} of a vertex $v\in V$ with respect to the partition $Π$ is the vector $r(v|Π)=(d(v,P_1),d(v,P_2),...,d(v,P_t))$, where $d(v,P_i)$ represents the distance between the vertex $v$ and the set $P_i$. A partition $Π$ of $V$ is a \emph{resolving partition} if different vertices of $G$ have different partition representations, i.e., for every pair of vertices $u,v\in V$, $r(u|Π)\ne r(v|Π)$. The \emph{partition dimension} of $G$ is the minimum number of sets in any resolving partition for $G$. In this paper we obtain several tight bounds on the partition dimension of unicyclic graphs.
The Chromatic Symmetric Function for Unicyclic Graphs
Universidad de Chile（智利大学）；Stephen F. Austin State University（斯蒂芬·F·奥斯汀州立大学）；Dartmouth College（达特茅斯学院）；Arizona State University（亚利桑那州立大学）；Willamette University（威拉米特大学）
The metric (resp. edge metric) dimension of a simple connected graph $G$, denoted by dim$(G)$ (resp. edim$(G)$), is the cardinality of a smallest vertex subset $S\subseteq V(G)$ for which every two distinct vertices (resp. edges) in $G$ have distinct distances to a vertex of $S$. It is an interesting topic to discuss the relation between dim$(G)$ and edim$(G)$ for some class of graphs $G$. In this paper, we settle two open problems on this topic for a widely studied class of graphs, called unicyclic graphs. Specifically, we introduce four classes of subgraphs to characterize the structure of a…
In a graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V (G) (resp. E(G)) is called the vertex (resp. edge) metric dimension of G. In [16] it was shown that both vertex and edge metric dimension of a unicyclic graph G always take values from just two explicitly given consecutive integers that are derived from the structure of the graph. A natural problem that arises is to determine under what conditions these dimensions take each of the two possible values. In this paper for each of these two metric dimensions we characterize three graph confi…
The \emph{metric dimension} $\dim(G)$ of a graph $G$ is the minimum number of vertices such that every vertex of $G$ is uniquely determined by its vector of distances to the chosen vertices. The \emph{zero forcing number} $Z(G)$ of a graph $G$ is the minimum cardinality of a set $S$ of black vertices (whereas vertices in $V(G)\!\setminus\!S$ are colored white) such that $V(G)$ is turned black after finitely many applications of "the color-change rule": a white vertex is converted black if it is the only white neighbor of a black vertex. We show that $\dim(T) \leq Z(T)$ for a tree $T$, and that…
In a graph G; a vertex (resp. an edge) metric generator is a set of vertices S such that any pair of vertices (resp. edges) from G is distinguished by at least one vertex from S: The cardinality of a smallest vertex (resp. edge) metric generator is the vertex (resp. edge) metric dimension of G: In [19] we determined the vertex (resp. edge) metric dimension of unicyclic graphs and that it takes its value from two consecutive integers. Therein, several cycle configurations were introduced and the vertex (resp. edge) metric dimension takes the greater of the two consecutive values only if any of …
In a graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V (G)[E(G) is called the mixed metric dimension of G. In this paper we first establish the exact value of the mixed metric dimension of a unicycic graph G which is derived from the structure of G. We further consider graphs G with edge disjoint cycles in which a unicyclic restriction Gi is introduced for each cycle Ci: Applying the result for unicyclic graph to each Gi then yields the exact value of the mixed metric dimension of such a graph G. The obtained formulas for the exact value of …
In a graph G, cardinality of the smallest ordered set of vertices that distinguishes every element of V (G) is the (vertex) metric dimension of G. Similarly, the cardinality of such a set is the edge metric dimension of G, if it distinguishes E(G). In this paper these invariants are considered first for unicyclic graphs, and it is shown that the vertex and edge metric dimensions obtain values from two particular consecutive integers, which can be determined from the structure of the graph. In particular, as a consequence, we obtain that these two invariants can differ for at most one for a sam…
The metric dimension dim(G) of a graph $G$ is the minimum cardinality of a subset $S$ of vertices of $G$ such that each vertex of $G$ is uniquely determined by its distances to $S$. It is well-known that the metric dimension of a graph can be drastically increased by the modification of a single edge. Our main result consists in proving that the increase of the metric dimension of an edge addition can be amortized in the sense that if the graph consists of a spanning tree $T$ plus $c$ edges, then the metric dimension of $G$ is at most the metric dimension of $T$ plus $6c$. We then use this res…
As a generalization of the concept of the partition dimension of a graph, this article introduces the notion of the $k$-partition dimension. Given a nontrivial connected graph $G=(V,E)$, a partition $Π$ of $V$ is said to be a $k$-partition generator for $G$ if any pair of different vertices $u,v\in V$ is distinguished by at least $k$ vertex sets of $Π$, \emph{i.e}., there exist at least $k$ vertex sets $S_1,\ldots,S_k\inΠ$ such that $d(u,S_i)\ne d(v,S_i)$ for every $i\in\{1,\ldots,k\}$. A $k$-partition generator for $G$ with minimum cardinality among all their $k$-partition generators is calle…
We characterize the downsets of integer partitions (ordered by containment of Ferrers diagrams) and compositions (ordered by the generalized subword order) which have finite dimension in the sense of Dushnik and Miller. In the case of partitions, while the set of all partitions has infinite dimension, we show that every proper downset of partitions has finite dimension. For compositions we identify four minimal downsets of infinite dimension and establish that every downset which does not contain one of these four has finite dimension.
Given an ordered partition $Π=\{P_1,P_2, ...,P_t\}$ of the vertex set $V$ of a connected graph $G=(V,E)$, the \emph{partition representation} of a vertex $v\in V$ with respect to the partition $Π$ is the vector $r(v|Π)=(d(v,P_1),d(v,P_2),...,d(v,P_t))$, where $d(v,P_i)$ represents the distance between the vertex $v$ and the set $P_i$. A partition $Π$ of $V$ is a \emph{resolving partition} of $G$ if different vertices of $G$ have different partition representations, i.e., for every pair of vertices $u,v\in V$, $r(u|Π)\ne r(v|Π)$. The \emph{partition dimension} of $G$ is the minimum number of se…
We determine the partition dimension of the incidence graph $G(Π_q)$ of the projective plane $Π_q$ up to a constant factor $2$ as $(2+o(1))\log_2{q}\leq \mathrm{pd}(G(Π_q))\leq (4+o(1))\log_2{q}.$
A partition P of the vertex set of a connected graph G is a locating partition of G if every vertex is uniquely determined by its vector of distances to the elements of P. The partition dimension of G is the minimum cardinality of a locating partition of G. A pair of vertices u,v of a graph G are called twins if they have exactly the same set of neighbors other than u and v. A twin class is a maximal set of pairwise twin vertices. The twin number of a graph G is the maximum cardinality of a twin class of G. In this paper we undertake the study of the partition dimension of a graph by also cons…
We construct a sequence of subset partition graphs satisfying the dimension reduction, adjacency, strong adjacency, and endpoint count properties whose diameter has a superlinear asymptotic lower bound. These abstractions of polytope graphs give further evidence against the Linear Hirsch Conjecture.
The notion of containment and avoidance provides a natural partial ordering on set partitions. Work of Sagan and of Goyt has led to enumerative results in avoidance classes of set partitions, which were refined by Dahlberg et al. through the use of combinatorial statistics. We continue this work by computing the distribution of the dimension index (a statistic arising from the supercharacter theory of finite groups) across certain avoidance classes of partitions. In doing so we obtain a novel connection between noncrossing partitions and 321-avoiding permutations, as well as connections to man…
A $d$-dimensional brick is a set $I_1\times \cdots \times I_d$ where each $I_i$ is an interval. Given a brick $B$, a brick partition of $B$ is a partition of $B$ into bricks. A brick partition $\mathcal{P}_d$ of a $d$-dimensional brick is $k$-piercing if every axis-parallel line intersects at least $k$ bricks in $\mathcal{P}_d$. Bucic et al. explicitly asked the minimum size $p(d, k)$ of a $k$-piercing brick partition of a $d$-dimensional brick. The answer is known to be $4(k-1)$ when $d=2$. Our first result almost determines $p(3, k)$. Namely, we construct a $k$-piercing brick partition of a …
We present a different way to obtain generators of metric spaces having the property that the ``position'' of every element of the space is uniquely determined by the distances from the elements of the generators. Specifically we introduce a generator based on a partition of the metric space into sets of elements. The sets of the partition will work as the new elements which will uniquely determine the position of each single element of the space. A set $W$ of vertices of a connected graph $G$ strongly resolves two different vertices $x,y\notin W$ if either $d_G(x,W)=d_G(x,y)+d_G(y,W)$ or $d_G…
The isometric path cover (partition) problem of a graph is to find a minimum set of isometric paths which cover (partition) the vertex set of the graph. The isometric path cover (partition) number of a graph is the cardinality a minimum isometric path cover (partition). We prove that the isometric path partition problem and the isometric $k$-path partition problem for $k\geq 3$ are NP-complete on general graphs. Fisher and Fitzpatrick \cite{FiFi01} have shown that the isometric path cover number of $(r\times r)$-dimensional grid is $\lceil 2r/3\rceil$. We show that the isometric path cover (pa…
A partition $Π=\{S_1,\ldots,S_k\}$ of the vertex set of a connected graph $G$ is called a \emph{resolving partition} of $G$ if for every pair of vertices $u$ and $v$, $d(u,S_j)\neq d(v,S_j)$, for some part $S_j$. The \emph{partition dimension} $β_p(G)$ is the minimum cardinality of a resolving partition of $G$. A resolving partition $Π$ is called \emph{resolving dominating} if for every vertex $v$ of $G$, $d(v,S_j)=1$, for some part $S_j$ of $Π$. The \emph{dominating partition dimension} $η_p(G)$ is the minimum cardinality of a resolving dominating partition of $G$. In this paper we show, amon…
A subset $S$ of the vertices $V$ of a connected graph $G$ resolves $G$ if no two vertices of $V$ share the same list of distances (shortest-path metric) with respect to the vertices of $S$ listed in a given order. The choice of such an $S$ in $V$ amounts to selecting a binary valued function $g$, said to be a resolving function, on $V$. The notion of a fractional resolving function is obtained by relaxing the codomain of $g$ to be the unit interval. Let $|g|=\sum_{v\in V}g(v)$. Given a finite collection $\mathcal{G}$ of connected graphs on a common vertex set $V$, the simultaneous metric dimen…
For a vertex $v$ of a connected graph $G(V,E)$ and a subset $S$ of $V$, the distance between $v$ and $S$ is defined by $d(v,S)=min\{d(v,x):x \in S \}.$ For an ordered \emph{k}-partition $Π=\{S_1,S_2\ldots S_k\}$ of $V$, the representation of $v$ with respect to $Π$ is the $k$-vector $r(v|Π) =(d(v,S_1),d(v,S_2)\ldots d(v,S_k)).$ The $k$-partition $Π$ is a resolving partition if the $k$-vectors $r(v|Π)$, $v \in V$ are distinct. The minimum $k$ for which there is a resolving $k$-partition of $V$ is the \emph{partition dimension} of $G$. Salman et al.{\rm\cite{SaJaCh12}} claimed that \emph{partiti…
Let $G=(V,E)$ be a connected graph. The distance between two vertices $u,v\in V$, denoted by $d(u, v)$, is the length of a shortest $u-v$ path in $G$. The distance between a vertex $v\in V$ and a subset $P\subset V$ is defined as $min\{d(v, x): x \in P\}$, and it is denoted by $d(v, P)$. An ordered partition $\{P_1,P_2, ...,P_t\}$ of vertices of a graph $G$, is a \emph{resolving partition}of $G$, if all the distance vectors $(d(v,P_1),d(v,P_2),...,d(v,P_t))$ are different. The \emph{partition dimension} of $G$, denoted by $pd(G)$, is the minimum number of sets in any resolving partition of $G$…
Some new necessary conditions for the existence of vector space partitions are derived. They are applied to the problem of finding the maximum number of spaces of dimension t in a vector space partition of V(2t,q) that contains m_d spaces of dimension d, where t/2<d<t, and also spaces of other dimensions. It is also discussed how this problem is related to maximal partial t-spreads in V(2t,q). We also give a lower bound for the number of spaces in a vector space partition and verify that this bound is tight.
An on-line chain partitioning algorithm receives a poset, one element at a time, and irrevocably assigns the element to one of the chains. Over 30 years ago, Szemerédi proved that any on-line algorithm could be forced to use $\binom{w+1}{2}$ chains to partition a poset of width $w$. The maximum number of chains that can be forced on any on-line algorithm remains unknown. In a survey paper by Bosek et al., it is shown that Szemerédi's argument could be improved to obtain a lower bound almost twice as good. Variants of the problem were considered where the class is restricted to posets of bounde…
Let $P$ be a set $n$ points in a $d$-dimensional space. Tverberg's theorem says that, if $n$ is at least $(k-1)(d+1)+1$, then $P$ can be partitioned into $k$ sets whose convex hulls intersect. Partitions with this property are called {\em Tverberg partitions}. A partition has tolerance $t$ if the partition remains a Tverberg partition after removal of any set of $t$ points from $P$. Tolerant Tverberg partitions exist in any dimension provided that $n$ is sufficiently large. Let $N(d,k,t)$ be the smallest value of $n$ such that tolerant Tverberg partitions exist for any set of $n$ points in $\m…
The \emph{zero forcing number}, $Z(G)$, of a graph $G$ is the minimum cardinality of a set $S$ of black vertices (whereas vertices in $V(G)-S$ are colored white) such that $V(G)$ is turned black after finitely many applications of "the color-change rule": a white vertex is converted black if it is the only white neighbor of a black vertex. The \emph{strong metric dimension}, $sdim(G)$, of a graph $G$ is the minimum among cardinalities of all strong resolving sets: $W \subseteq V(G)$ is a \emph{strong resolving set} of $G$ if for any $u, v \in V(G)$, there exists an $x \in W$ such that either $…
A vector space partition $\mathcal{P}$ in $\mathbb{F}_q^v$ is a set of subspaces such that every $1$-dimensional subspace of $\mathbb{F}_q^v$ is contained in exactly one element of $\mathcal{P}$. Replacing "every point" by "every $t$-dimensional subspace", we generalize this notion to vector space $t$-partitions and study their properties. There is a close connection to subspace codes and some problems are even interesting and unsolved for the set case $q=1$.
Algorithms and hardness for Metric Dimension on digraphs
Université Clermont Auvergne, CNRS, Clermont Auvergne INP, Mines Saint-Étienne, LIMOS（克莱蒙奥弗涅大学，法国国家科学研究中心，克莱蒙奥弗涅理工学院，圣艾蒂安国立高等矿业学校，LIMOS）；Department of Mathematics and Statistics, University of Turku（图尔库大学数学与统计系）；Turku Collegium for Science, Medicine and Technology TCSMT（图尔库科学技术医学学院）
Let $G$ be a connected graph with vertex set $V(G)$ and edge set $E(G)$. For an ordered $k$-partition $Π=\{Q_1,\ldots,Q_k\}$ of $V(G)$, the representation of a vertex $v \in V(G)$ with respect to $Π$ is the $k$-vectors $r(v|Π)=(d(v,Q_1),\ldots,d(v,Q_k))$, where $d(v,Q_i)$ is the distance between $v$ and $Q_i$. The partition $Π$ is a resolving partition if $r(u|Π)\neq r(v|Π)$, for each pair of distinct vertices $u,v \in V(G)$. The minimum $k$ for which there is a resolving $k$-partition of $V(G)$ is the partition dimension of $G$. A vertex $w\in V(G)$ strongly resolves two distinct vertices $u,…
We introduce the quasi-partition algebra $QP_k(n)$ as a centralizer algebra of the symmetric group. This algebra is a subalgebra of the partition algebra and inherits many similar combinatorial properties. We construct a basis for $QP_k(n)$, give a formula for its dimension in terms of the Bell numbers, and describe a set of generators for $QP_k(n)$ as a complex algebra. In addition, we give the dimensions and indexing set of its irreducible representations. We also provide the Bratteli diagram for the tower of quasi-partition algebras (constructed by letting $k$ range over the positive intege…
We highlight a topological aspect of the graph limit theory. Graphons are limit objects for convergent sequences of dense graphs. We introduce the representation of a graphon on a unique metric space and we relate the dimension of this metric space to the size of regularity partitions. We prove that if a graphon has an excluded induced sub-bigraph then the underlying metric space is compact and has finite packing dimension. It implies in particular that such graphons have regularity partitions of polynomial size.
We study generating functions of ordinary and plane partitions coloured by the action of a finite subgroup of the corresponding special linear group. After reviewing known results for the case of ordinary partitions, we formulate a conjecture concerning a factorisation property of the generating function of coloured plane partitions that can be thought of as an orbifold analogue of a conjecture of Maulik et al., now a theorem, in three-dimensional Donaldson-Thomas theory. We study natural quantisations of the generating functions arising from geometry, discuss a quantised version of our conjec…
