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We consider the following two algorithmic problems: given a graph $G$ and a subgraph $H\subseteq G$, decide whether $H$ is an isometric or a geodesically convex subgraph of $G$. It is relatively easy to see that the problems can be solved…
The mixed metric dimension ${\rm mdim}(G)$ of a graph $G$ is the cardinality of a smallest set of vertices that (metrically) resolves each pair of elements from $V(G)\cup E(G)$. We say that $G$ is a max-mdim graph if ${\rm mdim}(G) = n(G)$.…
Assume that $G$ is a graph with edge ideal $I(G)$. We provide sharp lower bounds for the depth of $I(G)^2$ in terms of the star packing number of $G$.
Let $G$ be a connected algebraic group. An unrefinable chain of $G$ is a chain of subgroups $G = G_0 > G_1 > \cdots > G_t = 1$, where each $G_i$ is a maximal connected subgroup of $G_{i-1}$. We introduce the notion of the length…
The depth of a link measures the minimum height of a resolving tree for the link whose leaves are all unlinks. We show that the depth of the closure of a strictly positive braid word is the length of the word minus the number of distinct…
Two graphs are co-spectral if their respective adjacency matrices have the same multi-set of eigenvalues. A graph is said to be determined by its spectrum if all graphs that are co-spectral with it are isomorphic to it. We consider these…
We define a special case of tree decompositions for planar graphs that respect a given embedding of the graph. We study the analogous width of the resulting decomposition we call the embedded-width of a plane graph. We show both upper…
A pair of non-adjacent edges is said to be separated in a circular ordering of vertices, if the endpoints of the two edges do not alternate in the ordering. The circular separation dimension of a graph $G$, denoted by $\pi^\circ(G)$, is the…
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 $G$…
The Weisfeiler-Leman (WL) dimension is an established measure for the inherent descriptive complexity of graphs and relational structures. It corresponds to the number of variables that are needed and sufficient to define the object of…
Suppose a finite, unweighted, combinatorial graph $G = (V,E)$ is the union of several (degree-)regular graphs which are then additionally connected with a few additional edges. $G$ will then have only a small number of vertices $v \in V$…
We describe a polynomial time algorithm for, given an undirected graph G, finding the minimum dimension d such that G may be isometrically embedded into the d-dimensional integer lattice Z^d.
This paper defines, for each graph $G$, a flag vector $fG$. The flag vectors of the graphs on $n$ vertices span a space whose dimension is $p(n)$, the number of partitions on $n$. The analogy with convex polytopes indicates that the linear…
A graph is reducible if it is the lexicographic product of two smaller non-trivial graphs. It is well-known a 1-planar graph with $n ~(\ge3)$ vertices has at most $4n-8$ edges, and a graph $G$ with $n$ vertices is optimal if $G$ has exactly…
We say that a first order formula A distinguishes a graph G from another graph G' if A is true on G and false on G'. Provided G and G' are non-isomorphic, let D(G,G') denote the minimal quantifier rank of a such formula. We prove that, if G…
The metric dimension of a graph is the least number of vertices in a set with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. Bailey and Meagher obtained an upper bound on the…
Let $G$ be a simple graph on $n$ vertices. We introduce the notion of bipartite connectivity of $G$, denoted by $\operatorname{bc}(G)$ and prove that $$\lim_{s \to \infty} \operatorname{depth} (S/I(G)^{(s)}) \le \operatorname{bc}(G),$$…
Two graph parameters are said to be coarsely equivalent if they are within constant factors from each other for every graph $G$. Recently, several graph parameters were shown to be coarsely equivalent to tree-length. Recall that the length…
For a finite binary string $x$ its logical depth $d$ for significance $b$ is the shortest running time of a program for $x$ of length $K(x)+b$. There is another definition of logical depth. We give a new proof that the two versions are…
A bisection of a graph is a bipartition of its vertex set such that the two resulting parts differ in size by at most 1, and its size is the number of edges that connect vertices in the two parts. The perfect matching condition and…