An Upper Bound on the Size of Obstructions for Bounded Linear Rank-Width

We provide a doubly exponential upper bound in $p$ on the size of forbidden pivot-minors for symmetric or skew-symmetric matrices over a fixed finite field $\mathbb{F}$ of linear rank-width at most $p$. As a corollary, we obtain a doubly exponential upper bound in $p$ on the size of forbidden vertex-minors for graphs of linear rank-width at most $p$. This solves an open question raised by Jeong, Kwon, and Oum [Excluded vertex-minors for graphs of linear rank-width at most $k$. European J. Combin., 41:242--257, 2014]. We also give a doubly exponential upper bound in $p$ on the size of forbidden minors for matroids representable over a fixed finite field of path-width at most $p$. Our basic tool is the pseudo-minor order used by Lagergren [Upper Bounds on the Size of Obstructions and Interwines, Journal of Combinatorial Theory Series B, 73:7--40, 1998] to bound the size of forbidden graph minors for bounded path-width. To adapt this notion into linear rank-width, it is necessary to well define partial pieces of graphs and merging operations that fit to pivot-minors. Using the algebraic operations introduced by Courcelle and Kant\'e, and then extended to (skew-)symmetric matrices by Kant\'e and Rao, we define boundaried $s$-labelled graphs and prove similar structure theorems for pivot-minor and linear rank-width.Comment: 28 pages, 1 figur

The average cut-rank of graphs

The cut-rank of a set $X$ of vertices in a graph $G$ is defined as the rank of the $X \times (V(G)\setminus X)$ matrix over the binary field whose $(i,j)$-entry is $1$ if the vertex $i$ in $X$ is adjacent to the vertex $j$ in $V(G)\setminus X$ and $0$ otherwise. We introduce the graph parameter called the average cut-rank of a graph, defined as the expected value of the cut-rank of a random set of vertices. We show that this parameter does not increase when taking vertex-minors of graphs and a class of graphs has bounded average cut-rank if and only if it has bounded neighborhood diversity. This allows us to deduce that for each real $\alpha$, the list of induced-subgraph-minimal graphs having average cut-rank larger than (or at least) $\alpha$ is finite. We further refine this by providing an upper bound on the size of obstruction and a lower bound on the number of obstructions for average cut-rank at most (or smaller than) $\alpha$ for each real $\alpha\ge0$. Finally, we describe explicitly all graphs of average cut-rank at most $3/2$ and determine up to $3/2$ all possible values that can be realized as the average cut-rank of some graph.Comment: 22 pages, 1 figure. The bound $x_n$ is corrected. Accepted to European J. Combinatoric

Compressive PCA for Low-Rank Matrices on Graphs

We introduce a novel framework for an approxi- mate recovery of data matrices which are low-rank on graphs, from sampled measurements. The rows and columns of such matrices belong to the span of the first few eigenvectors of the graphs constructed between their rows and columns. We leverage this property to recover the non-linear low-rank structures efficiently from sampled data measurements, with a low cost (linear in n). First, a Resrtricted Isometry Property (RIP) condition is introduced for efficient uniform sampling of the rows and columns of such matrices based on the cumulative coherence of graph eigenvectors. Secondly, a state-of-the-art fast low-rank recovery method is suggested for the sampled data. Finally, several efficient, parallel and parameter-free decoders are presented along with their theoretical analysis for decoding the low-rank and cluster indicators for the full data matrix. Thus, we overcome the computational limitations of the standard linear low-rank recovery methods for big datasets. Our method can also be seen as a major step towards efficient recovery of non- linear low-rank structures. For a matrix of size n X p, on a single core machine, our method gains a speed up of $p^2/k$ over Robust Principal Component Analysis (RPCA), where k << p is the subspace dimension. Numerically, we can recover a low-rank matrix of size 10304 X 1000, 100 times faster than Robust PCA

The F2\mathbb{F}_2-Rank and Size of Graphs

We consider the extremal family of graphs of order $2^n$ in which no two vertices have identical neighbourhoods, yet the adjacency matrix has rank only $n$ over the field of two elements. A previous result from algebraic geometry shows that such graphs exist for all even $n$ and do not exist for odd $n$. In this paper we provide a new combinatorial proof for this result, offering greater insight to the structure of graphs with these properties. We introduce a new graph product closely related to the Kronecker product, followed by a construction for such graphs for any even $n$. Moreover, we show that this is an infinite family of strongly-regular quasi-random graphs whose signed adjacency matrices are symmetric Hadamard matrices. Conversely, we provide a combinatorial proof that for all odd $n$, no twin-free graphs of minimal $\mathbb{F}_2$-rank exist, and that the next best-possible rank $(n+1)$ is attainable, which is tight.Comment: Added comparison to the results of Godsil and Royle. We thank Sam Adriaensen for bringing them to our attentio

Metric dimension of dual polar graphs

A resolving set for a graph $\Gamma$ is a collection of vertices $S$, chosen so that for each vertex $v$, the list of distances from $v$ to the members of $S$ uniquely specifies $v$. The metric dimension $\mu(\Gamma)$ is the smallest size of a resolving set for $\Gamma$. We consider the metric dimension of the dual polar graphs, and show that it is at most the rank over $\mathbb{R}$ of the incidence matrix of the corresponding polar space. We then compute this rank to give an explicit upper bound on the metric dimension of dual polar graphs.Comment: 8 page