| Summary: | This paper introduces a general approach to the idea of protection of graphs, which encompasses the known variants of secure domination and introduces new ones. Specifically, we introduce the study of secure <i>w</i>-domination in graphs, where <inline-formula><math display="inline"><semantics><mrow><mi>w</mi><mo>=</mo><mo>(</mo><msub><mi>w</mi><mn>0</mn></msub><mo>,</mo><msub><mi>w</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>w</mi><mi>l</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> is a vector of nonnegative integers such that <inline-formula><math display="inline"><semantics><mrow><msub><mi>w</mi><mn>0</mn></msub><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>. The secure <i>w</i>-domination number is defined as follows. Let <i>G</i> be a graph and <inline-formula><math display="inline"><semantics><mrow><mi>N</mi><mo>(</mo><mi>v</mi><mo>)</mo></mrow></semantics></math></inline-formula> the open neighborhood of <inline-formula><math display="inline"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>. We say that a function <inline-formula><math display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>⟶</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>l</mi><mo>}</mo></mrow></semantics></math></inline-formula> is a <i>w</i>-dominating function if <inline-formula><math display="inline"><semantics><mrow><mi>f</mi><mrow><mo>(</mo><mi>N</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>=</mo><msub><mo>∑</mo><mrow><mi>u</mi><mo>∈</mo><mi>N</mi><mo>(</mo><mi>v</mi><mo>)</mo></mrow></msub><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>≥</mo><msub><mi>w</mi><mi>i</mi></msub></mrow></semantics></math></inline-formula> for every vertex <i>v</i> with <inline-formula><math display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>=</mo><mi>i</mi></mrow></semantics></math></inline-formula>. The weight of <i>f</i> is defined to be <inline-formula><math display="inline"><semantics><mrow><mi>ω</mi><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mo>=</mo><msub><mo>∑</mo><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mi>f</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Given a <i>w</i>-dominating function <i>f</i> and any pair of adjacent vertices <inline-formula><math display="inline"><semantics><mrow><mi>v</mi><mo>,</mo><mi>u</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> with <inline-formula><math display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, the function <inline-formula><math display="inline"><semantics><msub><mi>f</mi><mrow><mi>u</mi><mo>→</mo><mi>v</mi></mrow></msub></semantics></math></inline-formula> is defined by <inline-formula><math display="inline"><semantics><mrow><msub><mi>f</mi><mrow><mi>u</mi><mo>→</mo><mi>v</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><msub><mi>f</mi><mrow><mi>u</mi><mo>→</mo><mi>v</mi></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><msub><mi>f</mi><mrow><mi>u</mi><mo>→</mo><mi>v</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for every <inline-formula><math display="inline"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>\</mo><mo>{</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>}</mo></mrow></semantics></math></inline-formula>. We say that a <i>w</i>-dominating function <i>f</i> is a secure <i>w</i>-dominating function if for every <i>v</i> with <inline-formula><math display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, there exists <inline-formula><math display="inline"><semantics><mrow><mi>u</mi><mo>∈</mo><mi>N</mi><mo>(</mo><mi>v</mi><mo>)</mo></mrow></semantics></math></inline-formula> such that <inline-formula><math display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msub><mi>f</mi><mrow><mi>u</mi><mo>→</mo><mi>v</mi></mrow></msub></semantics></math></inline-formula> is a <i>w</i>-dominating function as well. The secure <i>w</i>-domination number of <i>G</i>, denoted by <inline-formula><math display="inline"><semantics><mrow><msubsup><mi>γ</mi><mrow><mi>w</mi></mrow><mi>s</mi></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, is the minimum weight among all secure <i>w</i>-dominating functions. This paper provides fundamental results on <inline-formula><math display="inline"><semantics><mrow><msubsup><mi>γ</mi><mrow><mi>w</mi></mrow><mi>s</mi></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and raises the challenge of conducting a detailed study of the topic.
|
|---|