Certified domination critical graphs upon vertex removal
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Bibliographic record
Abstract
<p>A set <span class="math inline">\(D\)</span> of vertices of a graph <span class="math inline">\(G=(V_G,E_G)\)</span> is a <span><em>dominating set</em></span> of <span class="math inline">\(G\)</span> if every vertex in <span class="math inline">\(V_G-D\)</span> is adjacent to at least one vertex in <span class="math inline">\(D\)</span>. The <span><em>domination number</em></span> of a graph <span class="math inline">\(G\)</span>, denoted by <span class="math inline">\(\gamma(G)\)</span>, is the cardinality of a smallest dominating set of <span class="math inline">\(G\)</span>. A subset <span class="math inline">\(D\subseteq V_G\)</span> is called a <span><em>certified dominating set</em></span> of <span class="math inline">\(G\)</span> if <span class="math inline">\(D\)</span> is a dominating set of <span class="math inline">\(G\)</span>, and every vertex in <span class="math inline">\(D\)</span> has either zero or at least two neighbours in <span class="math inline">\(V_G-D\)</span>. The cardinality of a smallest certified dominating set of <span class="math inline">\(G\)</span> is called the <span><em>certified domination number</em></span> of <span class="math inline">\(G\)</span>, and it is denoted by <span class="math inline">\(\gamma_{\rm cer}(G)\)</span>. A vertex <span class="math inline">\(v\)</span> of <span class="math inline">\(G\)</span> is <em>certified critical</em> if <span class="math inline">\(\gamma_{\rm cer}(G -v) < \gamma_{\rm cer}(G)\)</span>, and a graph <span class="math inline">\(G\)</span> is <em>vertex certified domination critical</em> or <span class="math inline">\(\gamma_{cer}\)</span>-<em>critical</em> if the removal of any vertex reduces its certified domination number. In this paper, we give examples and properties of certified critical vertices and vertex certified domination critical graphs. As an example of an application of certified critical vertices, we give a constructive characterisation of trees for which the smaller partite set is a minimum certified dominating set.</p>
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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