COMPARING SMALL GRAPHS: HOW DISTANCE, ORIENTATION, AND ALIGNMENT AFFECT THE COMPARABILITY OF SMALL MULTIPLE BAR GRAPHS

Small multiples play a vital and growing role in the display of complex information. They are particularly useful for depicting spatiotemporal data, for which more traditional graphs and maps are inadequate. However, the scientific investigation of the usefulness of small multiples has been limited and often misdirected. In five experiments, small bar graphs are used to investigate several factors that could influence the comparability of the small graphs that comprise a small multiples graph. These factors include the distance between the graphs, the alignment of the graphs, the orientation of the bars, the length of the bars, and whether the graphs contain a single bar or multiple bars. In all cases, the most important factor affecting the comparability of the graphs was the difference in lengths, or difference in the increase of lengths, that the participants were asked to compare. The effects of distance were greater when the bars were closer to each other than when they were farther apart, suggesting that the bars are Title of Dissertation: COMPARING SMALL GRAPHS: HOW DISTANCE, ORIENTATION, AND ALIGNMENT AFFECT THE COMPARABILITY OF SMALL MULTIPLE BAR GRAPHS Benjamin Keniray Smith, Doctor of Philosophy, 2012 Dissertation Directed By: Professor Kent L. Norman, Department of Psychology compared using central vision. For pairs of graphs with a single bar each, comparability decreased as the distance between the graphs increased, although this effect was more prominent measured by accuracy than response time. Graph arrangements with horizontal alignments and vertical orientations were more comparable, although these effects were more subtle than the distance effects. For pairs of graphs with two bars each, the distance between the graphs had no effect on the accuracy of the comparison, and only a slight effect on the response time. Alignment and orientation had no effect on the comparability of graphs with two lines. The similarity of the lines in each graph, including but not limited to the critical length increase, significantly affected the comparability of the graphs. Part of a graph difficulty principle for small multiple graphs is offered as advice for graph creators

SING: Subgraph search In Non-homogeneous Graphs

<p>Abstract</p> <p>Background</p> <p>Finding the subgraphs of a graph database that are isomorphic to a given query graph has practical applications in several fields, from cheminformatics to image understanding. Since subgraph isomorphism is a computationally hard problem, indexing techniques have been intensively exploited to speed up the process. Such systems filter out those graphs which cannot contain the query, and apply a subgraph isomorphism algorithm to each residual candidate graph. The applicability of such systems is limited to databases of small graphs, because their filtering power degrades on large graphs.</p> <p>Results</p> <p>In this paper, SING (Subgraph search In Non-homogeneous Graphs), a novel indexing system able to cope with large graphs, is presented. The method uses the notion of <it>feature</it>, which can be a small subgraph, subtree or path. Each graph in the database is annotated with the set of all its features. The key point is to make use of feature locality information. This idea is used to both improve the filtering performance and speed up the subgraph isomorphism task.</p> <p>Conclusions</p> <p>Extensive tests on chemical compounds, biological networks and synthetic graphs show that the proposed system outperforms the most popular systems in query time over databases of medium and large graphs. Other specific tests show that the proposed system is effective for single large graphs.</p

Large cliques and independent sets all over the place

We study the following question raised by Erd\H{o}s and Hajnal in the early 90's. Over all $n$-vertex graphs $G$ what is the smallest possible value of $m$ for which any $m$ vertices of $G$ contain both a clique and an independent set of size $\log n$? We construct examples showing that $m$ is at most $2^{2^{(\log\log n)^{1/2+o(1)}}}$ obtaining a twofold sub-polynomial improvement over the upper bound of about $\sqrt{n}$ coming from the natural guess, the random graph. Our (probabilistic) construction gives rise to new examples of Ramsey graphs, which while having no very large homogenous subsets contain both cliques and independent sets of size $\log n$ in any small subset of vertices. This is very far from being true in random graphs. Our proofs are based on an interplay between taking lexicographic products and using randomness.Comment: 12 page

Crossing-number critical graphs have bounded path-width

AbstractThe crossing number of a graph G, denoted by cr(G), is defined as the smallest possible number of edge-crossings in a drawing of G in the plane. A graph G is crossing-critical if cr(G−e)<cr(G) for all edges e of G. We prove that crossing-critical graphs have “bounded path-width” (by a function of the crossing number), which roughly means that such graphs are made up of small pieces joined in a linear way on small cut-sets. Equivalently, a crossing-critical graph cannot contain a subdivision of a “large” binary tree. This assertion was conjectured earlier by Salazar (J. Geelen, B. Richter, G. Salazar, Embedding grids on surfaces, manuscript, 2000)

Extremal results on degree powers in some classes of graphs

Let $G$ be a simple graph of order $n$ with degree sequence $(d_1,d_2,\cdots,d_n)$. For an integer $p>1$, let $e_p(G)=\sum_{i=1}^n d^{p}_i$ and let $ex_p(n,H)$ be the maximum value of $e_p(G)$ among all graphs with $n$ vertices that do not contain $H$ as a subgraph (known as $H$-free graphs). Caro and Yuster proposed the problem of determining the exact value of $ex_2(n,C_4)$, where $C_4$ is the cycle of length $4$. In this paper, we show that if $G$ is a $C_4$-free graph having $n\geq 4$ vertices and $m\leq \lfloor 3(n-1)/2\rfloor$ edges and no isolated vertices, then $e_p(G)\leq e_p(F_n)$, with equality if and only if $G$ is the friendship graph $F_n$. This yields that for $n\geq 4$, $ex_p(n,\mathcal{C}^*)=e_p(F_n)$ and $F_n$ is the unique extremal graph, which is an improved complement of Caro and Yuster's result on $ex_p(n,\mathcal{C}^*)$, where $\mathcal{C}^*$ denotes the family of cycles of even lengths. We also determine the maximum value of $e_p(\cdot)$ among all minimally $t$-(edge)-connected graphs with small $t$ or among all $k$-degenerate graphs, and characterize the corresponding extremal graphs. A key tool in our approach is majorization