Domination and total domination in cubic graphs of large girth

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• 作者：Simone Dantas^a ; ^{sdantas@im.uff.br" class="auth_mail} ; Felix Joos^b ; ^{felix.joos@uni-ulm.de" class="auth_mail} ; Christian Lö ; wenstein^b ; ^{christian.loewenstein@uni-ulm.de" class="auth_mail} ; Deiwison S. Machado^a ; ^{dws.sousa@gmail.com" class="auth_mail} ; Dieter Rautenbach^b ; ^{dieter.rautenbach@uni-ulm.de" class="auth_mail} ; ^{dieter.rautenbach@googlemail.com" class="auth_mail}
• 关键词：Domination ; Total domination ; Cubic graph ; Girth
• 刊名：Discrete Applied Mathematics
• 出版年：10 September 2014
• 年：2014
• 卷：174
• 期：Complete
• 页码：128-132
• 全文大小：359 K

文摘

The domination number 65b2a00bdb7d4ede62" title="Click to view the MathML source">纬(G)$纬\left(G\right)$ and the total domination number 纬_t(G)${纬}_t\left(G\right)$ of a graph G$G$ without an isolated vertex are among the most well-studied parameters in graph theory. While the inequality 纬_t(G)≤2纬(G)${纬}_t\left(G\right)\le 2纬\left(G\right)$ is an almost immediate consequence of the definition, the extremal graphs for this inequality are not well understood. Furthermore, even very strong additional assumptions do not allow us to improve the inequality by much.

In the present paper we consider the relation of 纬(G)$纬\left(G\right)$ and 纬_t(G)${纬}_t\left(G\right)$ for cubic graphs G$G$ of large girth. Clearly, in this case 纬(G)$纬\left(G\right)$ is at least n(G)/4$n\left(G\right)/4$ where n(G)$n\left(G\right)$ is the order of G$G$. If 纬(G)$纬\left(G\right)$ is close to n(G)/4$n\left(G\right)/4$, then this forces a certain structure within b0582f655a3096d887690d" title="Click to view the MathML source">G$G$. We exploit this structure and prove an upper bound on 纬_t(G)${纬}_t\left(G\right)$, which depends on the value of 5b0b4c3a13" title="Click to view the MathML source">纬(G)$纬\left(G\right)$. As a consequence, we can considerably improve the inequality 纬_t(G)≤2纬(G)${纬}_t\left(G\right)\le 2纬\left(G\right)$ for cubic graphs of large girth.