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• W1956452579 abstract "Let G be a simple, undirected, finite graph with vertex set $V(G)$ and edge set $E(G)$. A k-dimensional box is a Cartesian product of closed intervals $[a_1,b_1]times [a_2,b_2]timescdotstimes [a_k,b_k]$. The boxicity of G, box$(G)$, is the minimum integer k such that G can be represented as the intersection graph of k-dimensional boxes; i.e., each vertex is mapped to a k-dimensional box and two vertices are adjacent in G if and only if their corresponding boxes intersect. Let $mathcal{P}=(S,P)$ be a poset, where S is the ground set and P is a reflexive, antisymmetric and transitive binary relation on S. The dimension of $mathcal{P}$, $dim(mathcal{P})$, is the minimum integer t such that P can be expressed as the intersection of t total orders. Let $G_{mathcal{P}}$ be the underlying comparability graph of $mathcal{P}$; i.e., S is the vertex set and two vertices are adjacent if and only if they are comparable in $mathcal{P}$. It is a well-known fact that posets with the same underlying comparability graph have the same dimension. The first result of this paper links the dimension of a poset to the boxicity of its underlying comparability graph. In particular, we show that for any poset $mathcal{P}$, box$(G_{mathcal{P}})/(chi(G_{mathcal{P}})-1) le dim(mathcal{P})le 2mbox{box}(G_{mathcal{P}})$, where $chi(G_{mathcal{P}})$ is the chromatic number of $G_{mathcal{P}}$ and $chi(G_{mathcal{P}})ne1$. It immediately follows that if $mathcal{P}$ is a height-2 poset, then box$(G_{mathcal{P}})le dim(mathcal{P})le 2mbox{box}(G_{mathcal{P}})$ since the underlying comparability graph of a height-2 poset is a bipartite graph. The second result of the paper relates the boxicity of a graph G with a natural partial order associated with the extended double cover of G, denoted as $G_c$: Note that $G_c$ is a bipartite graph with partite sets A and B which are copies of $V(G)$ such that, corresponding to every $uin V(G)$, there are two vertices $u_Ain A$ and $u_Bin B$ and ${u_A,v_B}$ is an edge in $G_c$ if and only if either $u=v$ or u is adjacent to v in G. Let $mathcal{P}_c$ be the natural height-2 poset associated with $G_c$ by making A the set of minimal elements and B the set of maximal elements. We show that $frac{mbox{box}(G)}{2} le dim(mathcal{P}_c) le 2mbox{box}(G)+4$. These results have some immediate and significant consequences. The upper bound $dim(mathcal{P})le 2mbox{box}(G_mathcal{P})$ allows us to derive hitherto unknown upper bounds for poset dimension such as $dim(mathcal{P})le 2mbox{ tree width }(G_{mathcal{P}})+4$, since boxicity of any graph is known to be at most its tree width $+; 2$. In the other direction, using the already known bounds for partial order dimension we get the following: (1) The boxicity of any graph with maximum degree $Delta$ is $O(Deltalog^2Delta)$, which is an improvement over the best-known upper bound of $Delta^2+2$. (2) There exist graphs with boxicity $Omega(DeltalogDelta)$. This disproves a conjecture that the boxicity of a graph is $O(Delta)$. (3) There exists no polynomial-time algorithm to approximate the boxicity of a bipartite graph on n vertices with a factor of $O(n^{0.5-epsilon})$ for any $epsilon>0$ unless $NP=ZPP$." @default.
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• W1956452579 date "2011-01-01" @default.
• W1956452579 modified "2023-09-25" @default.
• W1956452579 title "Boxicity and Poset Dimension" @default.
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• W1956452579 doi "https://doi.org/10.1137/100786290" @default.
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