On Metric Dimension of Functigraphs

The \emph{metric dimension} of a graph $G$, denoted by $\dim(G)$, is the minimum number of vertices such that each vertex is uniquely determined by its distances to the chosen vertices. Let $G_1$ and $G_2$ be disjoint copies of a graph $G$ and let $f: V(G_1) \rightarrow V(G_2)$ be a function. Then a \emph{functigraph} $C(G, f)=(V, E)$ has the vertex set $V=V(G_1) \cup V(G_2)$ and the edge set $E=E(G_1) \cup E(G_2) \cup \{uv \mid v=f(u)\}$. We study how metric dimension behaves in passing from $G$ to $C(G,f)$ by first showing that $2 \le \dim(C(G, f)) \le 2n-3$, if $G$ is a connected graph of order $n \ge 3$ and $f$ is any function. We further investigate the metric dimension of functigraphs on complete graphs and on cycles.Comment: 10 pages, 7 figure

Monotone Maps, Sphericity and Bounded Second Eigenvalue

We consider {\em monotone} embeddings of a finite metric space into low dimensional normed space. That is, embeddings that respect the order among the distances in the original space. Our main interest is in embeddings into Euclidean spaces. We observe that any metric on $n$ points can be embedded into $l_2^n$, while, (in a sense to be made precise later), for almost every $n$-point metric space, every monotone map must be into a space of dimension $\Omega(n)$. It becomes natural, then, to seek explicit constructions of metric spaces that cannot be monotonically embedded into spaces of sublinear dimension. To this end, we employ known results on {\em sphericity} of graphs, which suggest one example of such a metric space - that defined by a complete bipartitegraph. We prove that an $\delta n$-regular graph of order $n$, with bounded diameter has sphericity $\Omega(n/(\lambda_2+1))$, where $\lambda_2$ is the second largest eigenvalue of the adjacency matrix of the graph, and 0 < \delta \leq \half is constant. We also show that while random graphs have linear sphericity, there are {\em quasi-random} graphs of logarithmic sphericity. For the above bound to be linear, $\lambda_2$ must be constant. We show that if the second eigenvalue of an $n/2$-regular graph is bounded by a constant, then the graph is close to being complete bipartite. Namely, its adjacency matrix differs from that of a complete bipartite graph in only $o(n^2)$ entries. Furthermore, for any 0 < \delta < \half, and $\lambda_2$, there are only finitely many $\delta n$-regular graphs with second eigenvalue at most $\lambda_2$

The Knill Graph Dimension from Clique Cover

In this paper we prove that the recursive (Knill) dimension of the join of two graphs has a simple formula in terms of the dimensions of the component graphs: dim (G1 + G2) = 1 + dim G1 + dim G2. We use this formula to derive an expression for the Knill dimension of a graph from its minimum clique cover. A corollary of the formula is that a graph made of the arbitrary union of complete graphs KN of the same order KN will have dimension N − 1

Search for the end of a path in the d-dimensional grid and in other graphs

We consider the worst-case query complexity of some variants of certain \cl{PPAD}-complete search problems. Suppose we are given a graph $G$ and a vertex $s \in V(G)$. We denote the directed graph obtained from $G$ by directing all edges in both directions by $G'$. $D$ is a directed subgraph of $G'$ which is unknown to us, except that it consists of vertex-disjoint directed paths and cycles and one of the paths originates in $s$. Our goal is to find an endvertex of a path by using as few queries as possible. A query specifies a vertex $v\in V(G)$, and the answer is the set of the edges of $D$ incident to $v$, together with their directions. We also show lower bounds for the special case when $D$ consists of a single path. Our proofs use the theory of graph separators. Finally, we consider the case when the graph $G$ is a grid graph. In this case, using the connection with separators, we give asymptotically tight bounds as a function of the size of the grid, if the dimension of the grid is considered as fixed. In order to do this, we prove a separator theorem about grid graphs, which is interesting on its own right