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تسجيل مستخدم جديدUnit Disk Representations of Embedded Trees, Outerplanar and Multi-Legged Graphs
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A unit disk intersection representation (UDR) of a graph $G$ represents each vertex of $G$ as a unit disk in the plane, such that two disks intersect if and only if their vertices are adjacent in $G$. A UDR with interior-disjoint disks is called a unit disk contact representation (UDC). We prove that it is NP-hard to decide if an outerplanar graph or an embedded tree admits a UDR. We further provide a linear-time decidable characterization of caterpillar graphs that admit a UDR. Finally we show that it can be decided in linear time if a lobster graph admits a weak UDC, which permits intersections between disks of non-adjacent vertices.
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We prove a geometric version of the graph separator theorem for the unit disk intersection graph: for any set of $n$ unit disks in the plane there exists a line $ell$ such that $ell$ intersects at most $O(sqrt{(m+n)log{n}})$ disks and each of the hal
fplanes determined by $ell$ contains at most $2n/3$ unit disks from the set, where $m$ is the number of intersecting pairs of disks. We also show that an axis-parallel line intersecting $O(sqrt{m+n})$ disks exists, but each halfplane may contain up to $4n/5$ disks. We give an almost tight lower bound (up to sublogarithmic factors) for our approach, and also show that no line-separator of sublinear size in $n$ exists when we look at disks of arbitrary radii, even when $m=0$. Proofs are constructive and suggest simple algorithms that run in linear time. Experimental evaluation has also been conducted, which shows that for random instances our method outperforms the method by Fox and Pach (whose separator has size $O(sqrt{m})$).
Weak unit disk contact graphs are graphs that admit representing nodes as a collection of internally disjoint unit disks whose boundaries touch if there is an edge between the corresponding nodes. In this work we focus on graphs without embedding, i.
e., the neighbor order can be chosen arbitrarily. We give a linear time algorithm to recognize whether a caterpillar, a graph where every node is adjacent to or on a central path, allows a weak unit disk contact representation. On the other hand, we show that it is NP-hard to decide whether a tree allows such a representation.
Weak unit disk contact graphs are graphs that admit a representation of the nodes as a collection of internally disjoint unit disks whose boundaries touch if there is an edge between the corresponding nodes. We provide a gadget-based reduction to sho
w that recognizing embedded caterpillars that admit a weak unit disk contact representation is NP-hard.
Let $Vsubsetmathbb{R}^2$ be a set of $n$ sites in the plane. The unit disk graph $DG(V)$ of $V$ is the graph with vertex set $V$ in which two sites $v$ and $w$ are adjacent if and only if their Euclidean distance is at most $1$. We develop a compact
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Given a set P of n points in the plane, a unit-disk graph G_{r}(P) with respect to a radius r is an undirected graph whose vertex set is P such that an edge connects two points p, q in P if the Euclidean distance between p and q is at most r. The len
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