On some special classes of contact B0B_0-VPG graphs

A graph $G$ is a $B_0$-VPG graph if one can associate a path on a rectangular grid with each vertex such that two vertices are adjacent if and only if the corresponding paths intersect at at least one grid-point. A graph $G$ is a contact $B_0$-VPG graph if it is a $B_0$-VPG graph admitting a representation with no two paths crossing and no two paths sharing an edge of the grid. In this paper, we present a minimal forbidden induced subgraph characterisation of contact $B_0$-VPG graphs within four special graph classes: chordal graphs, tree-cographs, $P_4$-tidy graphs and $P_5$-free graphs. Moreover, we present a polynomial-time algorithm for recognising chordal contact $B_0$-VPG graphs.Comment: 34 pages, 15 figure

On contact graphs of paths on a grid

In this paper we consider Contact graphs of Paths on a Grid (CPG graphs), i.e. graphs for which there exists a family of interiorly disjoint paths on a grid in one-to-one correspondence with their vertex set such that two vertices are adjacent if and only if the corresponding paths touch at a grid-point. Our class generalizes the well studied class of VCPG graphs (see [1]). We examine CPG graphs from a structural point of view which leads to constant upper bounds on the clique number and the chromatic number. Moreover, we investigate the recognition and 3-colorability problems for B0- CPG, a subclass of CPG. We further show that CPG graphs are not necessarily planar and not all planar graphs are CPG

Characterising Chordal Contact: Bo-VPG Graphs

A graph G is a Bo- VPG graph if it is the vertex intersection graph of horizontal and vertical paths on a grid. A graph G is a contact Bo- VPG graph if the vertices can be represented by interiorly disjoint horizontal or vertical paths on a grid and two vertices are adjacent if and only if the corresponding paths touch. In this paper, we present a minimal forbidden induced subgraph characterisation of contact Bo-VPG graphs within the class of chordal graphs and provide a polynomial-time algorithm for recognising these graphs

A Note on Plus-Contacts, Rectangular Duals, and Box-Orthogonal Drawings

A plus-contact representation of a planar graph $G$ is called $c$-balanced if for every plus shape $+_v$, the number of other plus shapes incident to each arm of $+_v$ is at most $c \Delta +O(1)$, where $\Delta$ is the maximum degree of $G$. Although small values of $c$ have been achieved for a few subclasses of planar graphs (e.g., $2$- and $3$-trees), it is unknown whether $c$-balanced representations with $c<1$ exist for arbitrary planar graphs. In this paper we compute $(1/2)$-balanced plus-contact representations for all planar graphs that admit a rectangular dual. Our result implies that any graph with a rectangular dual has a 1-bend box-orthogonal drawings such that for each vertex $v$, the box representing $v$ is a square of side length $\frac{deg(v)}{2}+ O(1)$.Comment: A poster related to this research appeared at the 25th International Symposium on Graph Drawing & Network Visualization (GD 2017

Exploring, Engaging, Understanding in Museums

Patterns of accessibility through the space of the exhibition, connections or separations among spaces or exhibition elements, sequencing and grouping of elements, form our perceptions and shape our understanding. Through a review of several previous studies and the presentation of new work, this paper suggests that these patterns of movement form the basis of visitor understanding and that these effects can be deliberately controlled and elaborated through a closer examination of the influence of the visual and perceptual properties of an exhibition. Furthermore, it is argued that there is also a spatial discourse based on patterns of access and visibility that flows in its own right, although not entirely separate from the curatorial narrative

Pixel and Voxel Representations of Graphs

We study contact representations for graphs, which we call pixel representations in 2D and voxel representations in 3D. Our representations are based on the unit square grid whose cells we call pixels in 2D and voxels in 3D. Two pixels are adjacent if they share an edge, two voxels if they share a face. We call a connected set of pixels or voxels a blob. Given a graph, we represent its vertices by disjoint blobs such that two blobs contain adjacent pixels or voxels if and only if the corresponding vertices are adjacent. We are interested in the size of a representation, which is the number of pixels or voxels it consists of. We first show that finding minimum-size representations is NP-complete. Then, we bound representation sizes needed for certain graph classes. In 2D, we show that, for $k$-outerplanar graphs with $n$ vertices, $\Theta(kn)$ pixels are always sufficient and sometimes necessary. In particular, outerplanar graphs can be represented with a linear number of pixels, whereas general planar graphs sometimes need a quadratic number. In 3D, $\Theta(n^2)$ voxels are always sufficient and sometimes necessary for any $n$-vertex graph. We improve this bound to $\Theta(n\cdot \tau)$ for graphs of treewidth $\tau$ and to $O((g+1)^2n\log^2n)$ for graphs of genus $g$. In particular, planar graphs admit representations with $O(n\log^2n)$ voxels

Rectangular Layouts and Contact Graphs

Contact graphs of isothetic rectangles unify many concepts from applications including VLSI and architectural design, computational geometry, and GIS. Minimizing the area of their corresponding {\em rectangular layouts} is a key problem. We study the area-optimization problem and show that it is NP-hard to find a minimum-area rectangular layout of a given contact graph. We present O(n)-time algorithms that construct $O(n^2)$-area rectangular layouts for general contact graphs and $O(n\log n)$-area rectangular layouts for trees. (For trees, this is an $O(\log n)$-approximation algorithm.) We also present an infinite family of graphs (rsp., trees) that require $\Omega(n^2)$ (rsp., $\Omega(n\log n)$) area. We derive these results by presenting a new characterization of graphs that admit rectangular layouts using the related concept of {\em rectangular duals}. A corollary to our results relates the class of graphs that admit rectangular layouts to {\em rectangle of influence drawings}.Comment: 28 pages, 13 figures, 55 references, 1 appendi

Path, theme and narrative in open plan exhibition settings

Three arguments are made based on the analysis of science exhibitions. First,sufficiently refined techniques of spatial analysis allow us to model the impact oflayout upon visitors' paths, even in moderately sized open plans which allow almostrandom patterns of movement and relatively unobstructed visibility. Second, newlydeveloped or adapted techniques of analysis allow us to make a transition frommodeling the mechanics of spatial movement (the way in which movement is affectedby the distribution of obstacles and boundaries), to modeling the manner in whichmovement might register additional aspects of visual information. Third, theadvantages of such purely spatial modes of analysis extend into providing us with asharper understanding of some of the pragmatic constrains within which exhibitioncontent is conceived and designed