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Unit Disk Graph Recognition is NPHard
 Computational Geometry. Theory and Applications
, 1993
"... Unit disk graphs are the intersection graphs of unit diameter closed disks in the plane. This paper reduces SATISFIABILITY to the problem of recognizing unit disk graphs. Equivalently, it shows that determining if a graph has sphericity 2 or less, even if the graph is planar or is known to have s ..."
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Unit disk graphs are the intersection graphs of unit diameter closed disks in the plane. This paper reduces SATISFIABILITY to the problem of recognizing unit disk graphs. Equivalently, it shows that determining if a graph has sphericity 2 or less, even if the graph is planar or is known to have sphericity at most 3, is NPhard. We show how this reduction can be extended to 3 dimensions, thereby showing that unit sphere graph recognition, or determining if a graph has sphericity 3 or less, is also NPhard. We conjecture that Ksphericity is NPhard for all fixed K greater than 1. 1 Introduction A unit disk graph is the intersection graph of a set of unit diameter closed disks in the plane. That is, each vertex corresponds to a disk in the plane, and two vertices are adjacent in the graph if the corresponding disks intersect. The set of disks is said to realize the graph. Of course, the unit of distance is not critical, since the disks realize the same graph even if the coordina...
Cubicity of Interval Graphs and the Claw Number
, 2009
"... Let G(V, E) be a simple, undirected graph where V is the set of vertices and E is the set of edges. A bdimensional cube is a Cartesian product I1 × I2 × · · · × Ib, where each Ii is a closed interval of unit length on the real line. The cubicity of G, denoted by cub(G) is the minimum positive ..."
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Let G(V, E) be a simple, undirected graph where V is the set of vertices and E is the set of edges. A bdimensional cube is a Cartesian product I1 × I2 × · · · × Ib, where each Ii is a closed interval of unit length on the real line. The cubicity of G, denoted by cub(G) is the minimum positive integer b such that the vertices in G can be mapped to axis parallel bdimensional cubes in such a way that two vertices are adjacent in G if and only if their assigned cubes intersect. An interval graph is a graph that can be represented as the intersection of intervals on the real line i.e., the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. Suppose S(m) denotes a star graph on m + 1 nodes. We define claw number ψ(G) of the graph to be the largest positive integer m such that S(m) is an induced subgraph of G. It can be easily shown that the cubicity of any graph is at least ⌈log 2 ψ(G)⌉. In this paper, we show that, for an interval graph G ⌈log 2 ψ(G) ⌉ ≤ cub(G) ≤ ⌈log 2 ψ(G) ⌉ + 2. It is not clear whether the upper bound of ⌈log 2 ψ(G) ⌉ + 2 is tight: Till now we are unable to find any interval graph with cub(G)> ⌈log 2 ψ(G)⌉. We also show that, for an interval graph G, cub(G) ≤ ⌈log 2 α⌉, where α is the independence number of G. Therefore, in the special case of ψ(G) = α, cub(G) is exactly ⌈log 2 α⌉. The concept of cubicity can be generalized by considering boxes instead of cubes. A bdimensional box is a Cartesian product I1 ×I2 × · · ·×Ib, where each Ii is a closed interval on the real line. The boxicity of a graph, denoted box(G), is the minimum k such that G is the intersection graph of kdimensional boxes. It is clear that box(G) ≤ cub(G). From the above result, it follows that for any graph G, cub(G) ≤ box(G) ⌈log 2 α⌉.
On the Cubicity of Interval Graphs
"... A kcube (or “a unit cube in k dimensions”) is defined as the Cartesian product R1 ×... × Rk where Ri(for 1 ≤ i ≤ k) is an interval of the form [ai, ai + 1] on the real line. The kcube representation of a graph G is a mapping of the vertices of G to kcubes such that the kcubes mapped to two vert ..."
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A kcube (or “a unit cube in k dimensions”) is defined as the Cartesian product R1 ×... × Rk where Ri(for 1 ≤ i ≤ k) is an interval of the form [ai, ai + 1] on the real line. The kcube representation of a graph G is a mapping of the vertices of G to kcubes such that the kcubes mapped to two vertices in G have a nonempty intersection if and only if the vertices are adjacent. The cubicity of a graph G, denoted as cub(G), is defined as the minimum dimension k such that G has a kcube representation. An interval graph is a graph that can be represented as the intersection of intervals on the real line i.e., the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. We show that for any interval graph G with maximum degree ∆, cub(G) ≤ ⌈log ∆ ⌉ + 4. This upper bound is shown to be tight up to an additive constant of 4 by demonstrating interval graphs for which cubicity is equal to ⌈log ∆⌉.
ALGEBRAIC DISTANCE GRAPHS AND RIGIDITY
"... Abstract. An algebraic distance graph is defined to be a graph with vertices in En in which two vertices are adjacent if and only if the distance between them is an algebraic number. It is proved that an algebraic distance graph with finite vertex set is complete if and only if the graph is "rigid". ..."
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Abstract. An algebraic distance graph is defined to be a graph with vertices in En in which two vertices are adjacent if and only if the distance between them is an algebraic number. It is proved that an algebraic distance graph with finite vertex set is complete if and only if the graph is "rigid". Applying this result, we prove that ( 1) if all the sides of a convex polygon T which is inscribed in a circle are algebraic numbers, then the circumradius and all diagonals of Y are also algebraic numbers, (2) the chromatic number of the algebraic distance graph on a circle of radius r is oo or 2 accordingly as r is algebraic or not. We also prove that for any n> 0, there exists a graph G which cannot be represented as an algebraic distance graph in En. 1.