Tyshkevich的图分解和单图的区分数

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics Pub Date : 2025-03-24 DOI:10.1016/j.disc.2025.114492

Christine T. Cheng

引用次数: 0

摘要

图G的一个c标记φ:V(G)→{1,2，…，c}是为了区分对于G的每一个非平凡自同构π，是否存在某个顶点V使得φ (V)≠φ (π(V))。G的区分数D(G)是最小的c，使得G有一个区分的c标记。我们考虑了一个紧凑版的Tyshkevich图分解定理，其中平凡分量被极大地组合以形成完全图或孤立顶点图。假设G的紧形正则分解是Gk°Gk−1°⋯G1°G0。我们证明了当且仅当当i=0，…，k时，当φ被限制为V（Gi）时，φ是Gi的一个可分辨标记。因此,D (G) = max⁡{D (Gi), i = 0,…,k}。然后，我们提出了一种在线性时间内计算单图的区分数的算法。

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Tyshkevich's graph decomposition and the distinguishing numbers of unigraphs

A c-labeling

ϕ : V (G) \to {1, 2, \dots, c}

of graph G is distinguishing if, for every non-trivial automorphism π of G, there is some vertex v so that

ϕ (v) \neq ϕ (π (v))

. The distinguishing number of G,

D (G)

, is the smallest c such that G has a distinguishing c-labeling.

We consider a compact version of Tyshkevich's graph decomposition theorem where trivial components are maximally combined to form a complete graph or a graph of isolated vertices. Suppose the compact canonical decomposition of G is

G_{k} \circ G_{k - 1} \circ \dots \circ G_{1} \circ G_{0}

. We prove that ϕ is a distinguishing labeling of G if and only if ϕ is a distinguishing labeling of

G_{i}

when restricted to

V (G_{i})

for

i = 0, \dots, k

. Thus,

D (G) = \max {D (G_{i}), i = 0, \dots, k}

. We then present an algorithm that computes the distinguishing number of a unigraph in linear time.

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来源期刊

Discrete Mathematics 数学-数学

CiteScore

1.50

自引率

12.50%

发文量

424

审稿时长

6 months

期刊介绍： Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.