Bandwidth and pathwidth of three-dimensional grids

In this paper, we study two well-known graph parameters, the bandwidth and the pathwidth. These two parameters were defined in different areas of Computer Science. However, there is a close relation between the two parameters. In fact, it is known that the bandwidth of a graph is at least its pathwidth. Furthermore, as we will show, these two parameters are identical for grids. Since gridlike graphs, especially two or three-dimensional grids, arise in many practical situations, several graph parameters of them have been studied intensively [24,25,26,5,17,1,21]. In particular, the bandwidth and the pathwidth of grids and tori were studied by several researchers [12,8,20,11]. However, closed formulas of these parameter for noncubic three-dimensional grids and four or more-dimensional grids were not known previously. We study these grids and present closed formulas for some cases.

The rest of this paper is organized as follows. In Section 2, we give some definitions and known results. In Section 3, we determine the two parameters for multi-dimensional grids that have relatively large maximum factors. Generally speaking, large-dimensional cases are difficult to handle. However, we show that if the maximum factor in a grid is relatively large then the two parameters can be easily determined. In Section 4, we determine the two parameters of threedimensional grids. FitzGerald [12] determined the bandwidth of cubic grids P n P n P n . We properly extend this result to noncubic cases P n 1 P n 2 P n 3 . In the last section, we conclude this paper and give a conjecture for the four-dimensional case.

In this section, we define some graph parameters, and a graph operation. Grids, and tori are also defined here. After the definitions, we provide some known results as well as some useful observations. All graphs in this paper are finite, simple, and connected. We denote the vertex and edge sets of a graph G by V(G) and E(G), respectively.

The bandwidth of graphs was defined by Harper [13]. An ordering of a graph G is a bijection

The bandwidth problem appears in a lot of areas of Computer Science such as VLSI layouts and parallel computing. See surveys [7,9].

The pathwidth of graphs was defined by Robertson and Seymour [23] in their work of the Graph Minor Theory. Given a graph G, a sequence X 1 , . . . , X r of subsets of V(G) is a path decomposition of G if the following conditions are satisfied:

The width of a path decomposition X 1 , . . . , X r is max 1≤i≤r |X i | -1. The pathwidth of G, denoted by pw(G), is the minimum width over all path decompositions of G.

The proper pathwidth of G, ppw(G), is the minimum width over all proper path decompositions of G. Clearly, pw(G) ≤ ppw(G) for any graph G. A nontrivial relation pw(G) ≤ bw(G) is a corollary of the following fact.

We define the vertex boundary width of G as vbw(G) = max 1≤k≤|V(G)| β G (k). We often omit the subscript G of ∂ G and β G if the graph G is clear from the context. The following theorem implies vbw(G) ≤ pw(G) for any graph G.

From the above observations, we have the inequality vbw(G) ≤ pw(G) ≤ bw(G) for any graph G. Harper [13,14] showed that the equality also holds for some graphs. An ordering on

for all k, where I k is the set of the first k vertices of V(G) in the ordering.

The observations in this subsection give the following corollary.

The Cartesian product of graphs G and H, denoted by G H, is the graph whose vertex set is V(G) × V(H) and in which a vertex (g, h) is adjacent to a vertex (g ′ , h ′ ) if and only if either g = g ′ and {h, h ′ } ∈ E(H), or h = h ′ and {g, g ′ } ∈ E(G). It is easy to see that the Cartesian product operation is associative and commutative up to isomorphism. We denote the Cartesian product of

We define the simplicial order ≺ on V( d i=1 P n i ) by setting (u 1 , . . . , u d ) ≺ (v 1 , . . . , v d ) if and only if either wei(u) < wei(v), or wei(u) = wei(v) and there exists an index j such that u j > v j and u i = v i for all i < j. Intuitively, vertices are ordered in the simplicial order by increasing weight and anti-lexicographically with each weight class [27]. For example, the vertices of P 2 P 3 P 3 are ordered as follows: (0, 0, 0) ≺ (1, 0, 0) ≺ (0, 1, 0) ≺ (0, 0, 1)

We also define , ≻, and naturally. Moghadam [18,19] and Bollobás and Leader [3,4] showed independently that the simplicial order is isoperimetric for grids. Theorem 2.5 ([3, 4, 18, 19]

Riordan [22] showed that even tori have an isoperimetric order. Thus, we also have the following equivalence.

However, we do not need to give a formal definition of Riordan’s ordering. This is because of the equivalence of the problem on even tori and grids. Recently, Bezrukov and Leck [2] have proved that the VIP on d-dimensional even tori is equivalent to the VIP on some 2d-dimensional grids.

From the above observations, the problems of determining the bandwidth and the pathwidth of grids and tori are solvable by determining the vertex boundary width

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