Embeddings of complete binary trees into grids and extended grids with total vertex-congestion 1

Abstract

Let G and H be two simple, undirected graphs. An embedding of the graph G into the graph H is an injective mapping f from the vertices of G to the vertices of H, together with a mapping which assigns to each edge [ u, v] of G a path between f( u) and f( v) in H. The grid M( r, s) is the graph whose vertex set is the set of pairs on nonnegative integers, {(i,j) : 0⩽i<r, 0⩽j<s} , in which there is an edge between vertices ( i, j) and ( k, l) if either | i− k|=1 and j= l or i= k and | j− l|=1. The extended grid EM( r, s) is the graph whose vertex set is the set of pairs on nonnegative integers, {(i,j) : 0⩽i<r, 0⩽j<s} , in which there is an edge between vertices ( i, j) and ( k, l) if and only if | i− k|⩽1 and | j− l|⩽1. In this paper, we give embeddings of complete binary trees into square grids and extended grids with total vertex-congestion 1, i.e., for any vertex x of the extended grid we have load( x)+ vertex- congestion( x)⩽1. Depending on the parity of the height of the tree, the expansion of these embeddings is approaching 1.606 or 1.51 for grids, and 1.208 or 1.247 for extended grids.