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arxiv: 2204.12079 · v4 · submitted 2022-04-26 · 💻 cs.DM

Exact Wirelength of Embedding 3-Ary n-Cubes into certain Cylinders and Trees

Rajeshwari S , M Rajesh This is my paper

Pith reviewed 2026-05-24 12:16 UTC · model grok-4.3

classification 💻 cs.DM
keywords graph embeddingwirelength3-ary n-cubecylinder graphtree graphinterconnection networksparallel algorithms
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The pith

The minimum wirelength for one-to-one embeddings of 3-ary n-cubes into cylinders and certain trees is determined exactly.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper calculates the exact minimum wirelength needed to map the vertices of a 3-ary n-cube onto the vertices of cylinder graphs and selected tree graphs. Wirelength is the total cost obtained by adding the host-graph distance for every edge of the guest graph. Lower wirelength corresponds to cheaper communication when one parallel architecture is simulated on another. The authors supply explicit constructions that reach these minimum totals for the chosen host graphs.

Core claim

The wirelength of an embedding of the 3-ary n-cube into cylinders and certain trees equals the value achieved by the given one-to-one vertex mappings that minimize the sum of distances in the host graph over all edges of the guest graph.

What carries the argument

Wirelength, defined as the sum over every guest edge of the distance between its two endpoints in the host graph, with the minimum taken over all possible one-to-one mappings.

Load-bearing premise

The given constructions are the ones that produce the smallest possible summed host distances for the selected cylinders and trees.

What would settle it

An explicit embedding of a 3-ary n-cube into one of the cylinders or trees whose total summed host distances is strictly smaller than the value reported in the paper.

Figures

Figures reproduced from arXiv: 2204.12079 by M Rajesh, Rajeshwari S.

Figure 1
Figure 1. Figure 1: 3-ary 3-cube, Q3 3 . Definition 3.2. [37] The Lexicographic order on a set of n-tuples with integer entries is defined as follows: We say that (x1, ..., xn) is greater than (y1, ..., yn) if there exist an index i, 1 ≤ i ≤ n, such that xj = yj for 1 ≤ j < i and xi > yi . Sergei et al. [37] has studied the edge isoperimetric problem for the torus C3 × C3 which was solved in [38, 39] by introducing a new char… view at source ↗
Figure 2
Figure 2. Figure 2: (a) Vertical edge cuts Xt i , 1 ≤ i ≤ 8, 1 ≤ t ≤ 2 of Cylinder C3 × P9 with lexicographic ordering. (b)Horizontal edge cuts Yj , 1 ≤ j ≤ 2 of Cylinder C3 × P9 with lexicographic ordering. Proof: Consider the lexicographic embedding lex : Q3 n → C3 × P3n−1 given in the Embedding Algorithm A. Xt i , i = 1, 2, .., 3 n−1 − 1 and t = 1, 2, shown in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Edge cuts of Caterpillar. Proof: By Theorem 3.3, the lexicographic ordering of vertices of Q3 n gives the optimal order for inducing the maximum subgraph. The edge cut Si removes the edges in the backbone of the caterpillar, such that each Si disconnects it into two components of lexicographic ordering which induce a maximum subgraph in Q3 n . The edge cut Tj disconnects the caterpillar with exactly one ve… view at source ↗
Figure 4
Figure 4. Figure 4: Edge cuts of F3n−1,3. Proof: The removal of edges in Si , 1 ≤ i ≤ 3 n−1−1 disconnects F3n−1,3 into two components whose inverse images under lex induce lexicographic ordering of the corresponding subgraphs of Q3 n . This implies that the inverse images are maximum subgraphs of Q3 n . ut [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Edge cuts of B 2,  3n 2 . Lemma 5.12. The edge cuts S 1 i and S 2 i , ∀ i = 1, 2, ...,  3 n 2  − 3 of B 2,  3n 2  as shown in [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
read the original abstract

Graph embeddings play a significant role in the design and analysis of parallel algorithms. It is a mapping of the topological structure of a guest graph G into a host graph H, which is represented as a one-to-one mapping from the vertex set of the guest graph to the vertex set of the host graph. In multiprocessing systems the interconnection networks enhance the efficient communication between the components in the system. Obtaining minimum wirelength in embedding problems is significant in the designing of network and simulating one architecture by another. In this paper, we determine the wirelength of embedding 3-ary n-cubes into cylinders and certain trees.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper claims to determine the exact wirelength of one-to-one embeddings of the 3-ary n-cube into cylinder graphs and certain tree graphs, where wirelength is defined as the minimum, over all such mappings, of the sum of host-graph distances realized by the guest-graph edges.

Significance. If the claimed exact values are correctly established by matching lower bounds and explicit constructions, the results supply precise quantitative data on embedding costs for a standard family of interconnection networks, which can inform the design and simulation of parallel architectures.

minor comments (3)
  1. [Abstract] The abstract states the main result but does not name the specific cylinder parameters or tree families for which exact wirelength is obtained; adding one sentence with these details would improve clarity.
  2. [Preliminaries] Notation for the 3-ary n-cube (e.g., Q_n^3) and the cylinder graphs should be introduced once in a dedicated preliminaries section and used consistently thereafter.
  3. [Figures] Figure captions for any embedding diagrams should explicitly state the guest and host graphs depicted and the value of n shown.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our results on exact wirelength for 3-ary n-cube embeddings and for recommending minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper determines exact wirelength for embeddings of 3-ary n-cubes into cylinders and trees via one-to-one vertex mappings and summed host distances. The abstract and description indicate standard lower-bound plus matching construction approach with no equations shown that reduce by definition to inputs, no fitted parameters renamed as predictions, and no load-bearing self-citations or ansatzes. The derivation is self-contained against the external definition of wirelength and the explicit constructions provided.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5628 in / 943 out tokens · 37964 ms · 2026-05-24T12:16:42.803830+00:00 · methodology

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