Computing the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"><a:mfenced open="(" close=")" separators="|"><a:mrow><a:mi>l</a:mi><a:mo>,</a:mo><a:mi>k</a:mi></a:mrow></a:mfenced></a:math>-Clique Metric Dimension of Graphs via (Edge) Corona Products and Integer Linear Programming Model

Let <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M2"><a:mi>G</a:mi></a:math> be a graph with <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M3"><c:mi>n</c:mi></c:math> vertices and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M4"><e:msub><e:mrow><e:mi>C</e:mi></e:mrow><e:mrow><e:mi>G</e:mi></e:mrow></e:msub><e:mo>=</e:mo><e:mfenced open="{" close="" separators="|"><e:mrow><e:mi>X</e:mi><e:mo>:</e:mo><e:mi>X</e:mi></e:mrow></e:mfenced></e:math> is an <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" id="M5"><j:mi>l</j:mi></j:math> -clique of <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" id="M6"><l:mfenced open="" close="}" separators="|"><l:mrow><l:mi>G</l:mi></l:mrow></l:mfenced></l:math> . A vertex <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" id="M7"><q:mi>v</q:mi><q:mo>∈</q:mo><q:mi>V</q:mi><q:mfenced open="(" close=")" separators="|"><q:mrow><q:mi>G</q:mi></q:mrow></q:mfenced></q:math> is said to resolve a pair of cliques <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" id="M8"><v:mfenced open="{" close="}" separators="|"><v:mrow><v:mi>X</v:mi><v:mo>,</v:mo><v:mi>Y</v:mi></v:mrow></v:mfenced></v:math> in <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" id="M9"><ab:mi>G</ab:mi></ab:math> if <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" id="M10"><cb:msub><cb:mrow><cb:mi>d</cb:mi></cb:mrow><cb:mrow><cb:mi>G</cb:mi></cb:mrow></cb:msub><cb:mfenced open="(" close=")" separators="|"><cb:mrow><cb:mi>v</cb:mi><cb:mo>,</cb:mo><cb:mi>X</cb:mi></cb:mrow></cb:mfenced><cb:mo>≠</cb:mo><cb:msub><cb:mrow><cb:mi>d</cb:mi></cb:mrow><cb:mrow><cb:mi>G</cb:mi></cb:mrow></cb:msub><cb:mrow><cb:mfenced open="(" close=")" separators="|"><cb:mrow><cb:mi>v</cb:mi><cb:mo>,</cb:mo><cb:mi>Y</cb:mi></cb:mrow></cb:mfenced></cb:mrow></cb:math> where <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" id="M11"><kb:msub><kb:mrow><kb:mi>d</kb:mi></kb:mrow><kb:mrow><kb:mi>G</kb:mi></kb:mrow></kb:msub></kb:math> is the distance function of <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" id="M12"><mb:mi>G</mb:mi></mb:math> . For a pair of cliques <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" id="M13"><ob:mfenced open="{" close="}" separators="|"><ob:mrow><ob:mi>X</ob:mi><ob:mo>,</ob:mo><ob:mi>Y</ob:mi></ob:mrow></ob:mfenced></ob:math> , the resolving neighbourhood of <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" id="M14"><tb:mi>X</tb:mi></tb:math> and <vb:math xmlns:vb="http://www.w3.org/1998/Math/MathML" id="M15"><vb:mi>Y</vb:mi></vb:math> , denoted by <xb:math xmlns:xb="http://www.w3.org/1998/Math/MathML" id="M16"><xb:msub><xb:mrow><xb:mi>R</xb:mi></xb:mrow><xb:mrow><xb:mi>G</xb:mi></xb:mrow></xb:msub><xb:mfenced open="{" close="}" separators="|"><xb:mrow><xb:mi>X</xb:mi><xb:mo>,</xb:mo><xb:mi>Y</xb:mi></xb:mrow></xb:mfenced></xb:math> , is the collection of all vertices which resolve the pair <cc:math xmlns:cc="http://www.w3.org/1998/Math/MathML" id="M17"><cc:mfenced open="{" close="}" separators="|"><cc:mrow><cc:mi>X</cc:mi><cc:mo>,</cc:mo><cc:mi>Y</cc:mi></cc:mrow></cc:mfenced></cc:math> . A subset <hc:math xmlns:hc="http://www.w3.org/1998/Math/MathML" id="M18"><hc:mi>S</hc:mi></hc:math> of <jc:math xmlns:jc="http://www.w3.org/1998/Math/MathML" id="M19"><jc:mi>V</jc:mi><jc:mfenced open="(" close=")" separators="|"><jc:mrow><jc:mi>G</jc:mi></jc:mrow></jc:mfenced></jc:math> is called an <oc:math xmlns:oc="http://www.w3.org/1998/Math/MathML" id="M20"><oc:mfenced open="(" close=")" separators="|"><oc:mrow><oc:mi>l</oc:mi><oc:mo>,</oc:mo><oc:mi>k</oc:mi></oc:mrow></oc:mfenced></oc:math> -clique metric generator for <tc:math xmlns:tc="http://www.w3.org/1998/Math/MathML" id="M21"><tc:mi>G</tc:mi></tc:math> if <vc:math xmlns:vc="http://www.w3.org/1998/Math/MathML" id="M22"><vc:mfenced open="|" close="|" separators="|"><vc:mrow><vc:msub><vc:mrow><vc:mi>R</vc:mi></vc:mrow><vc:mrow><vc:mi>G</vc:mi></vc:mrow></vc:msub><vc:mrow><vc:mfenced open="{" close="}" separators="|"><vc:mrow><vc:mi>X</vc:mi><vc:mo>,</vc:mo><vc:mi>Y</vc:mi></vc:mrow></vc:mfenced></vc:mrow><vc:mo>∩</vc:mo><vc:mi>S</vc:mi></vc:mrow></vc:mfenced><vc:mo>≥</vc:mo><vc:mi>k</vc:mi></vc:math> for each pair of distinct <dd:math xmlns:dd="http://www.w3.org/1998/Math/MathML" id="M23"><dd:mi>l</dd:mi></dd:math> -cliques <fd:math xmlns:fd="http://www.w3.org/1998/Math/MathML" id="M24"><fd:mi>X</fd:mi></fd:math> and <hd:math xmlns:hd="http://www.w3.org/1998/Math/MathML" id="M25"><hd:mi>Y</hd:mi></hd:math> of <jd:math xmlns:jd="http://www.w3.org/1998/Math/MathML" id="M26"><jd:mi>G</jd:mi></jd:math> . The <ld:math xmlns:ld="http://www.w3.org/1998/Math/MathML" id="M27"><ld:mfenced open="(" close=")" separators="|"><ld:mrow><ld:mi>l</ld:mi><ld:mo>,</ld:mo><ld:mi>k</ld:mi></ld:mrow></ld:mfenced></ld:math> -clique metric dimension of <qd:math xmlns:qd="http://www.w3.org/1998/Math/MathML" id="M28"><qd:mi>G</qd:mi></qd:math> , denoted by <sd:math xmlns:sd="http://www.w3.org/1998/Math/MathML" id="M29"><sd:mi>l</sd:mi><sd:mo>−</sd:mo><sd:msub><sd:mrow><sd:mtext>cdim</sd:mtext></sd:mrow><sd:mrow><sd:mi>k</sd:mi></sd:mrow></sd:msub><sd:mfenced open="(" close=")" separators="|"><sd:mrow><sd:mi>G</sd:mi></sd:mrow></sd:mfenced></sd:math> , is defined as <xd:math xmlns:xd="http://www.w3.org/1998/Math/MathML" id="M30"><xd:mi mathvariant="normal">min</xd:mi><xd:mfenced open="{" close="" separators="|"><xd:mrow><xd:mrow><xd:mfenced open="|" close="|" separators="|"><xd:mrow><xd:mi>S</xd:mi></xd:mrow></xd:mfenced></xd:mrow><xd:mo>:</xd:mo><xd:mi>S</xd:mi></xd:mrow></xd:mfenced></xd:math> is an <ge:math xmlns:ge="http://www.w3.org/1998/Math/MathML" id="M31"><ge:mfenced open="(" close=")" separators="|"><ge:mrow><ge:mi>l</ge:mi><ge:mo>,</ge:mo><ge:mi>k</ge:mi></ge:mrow></ge:mfenced></ge:math> -clique metric generator of <le:math xmlns:le="http://www.w3.org/1998/Math/MathML" id="M32"><le:mfenced open="" close="}" separators="|"><le:mrow><le:mi>G</le:mi></le:mrow></le:mfenced></le:math> . In this paper, the <qe:math xmlns:qe="http://www.w3.org/1998/Math/MathML" id="M33"><qe:mfenced open="(" close=")" separators="|"><qe:mrow><qe:mi>l</qe:mi><qe:mo>,</qe:mo><qe:mi>k</qe:mi></qe:mrow></qe:mfenced></qe:math> -clique metric dimension of corona and edge corona of two graphs are computed. In addition, an integer linear programming model is presented for the <ve:math xmlns:ve="http://www.w3.org/1998/Math/MathML" id="M34"><ve:mfenced open="(" close=")" separators="|"><ve:mrow><ve:mi>l</ve:mi><ve:mo>,</ve:mo><ve:mi>k</ve:mi></ve:mrow></ve:mfenced></ve:math> -clique metric basis for a given graph <af:math xmlns:af="http://www.w3.org/1998/Math/MathML" id="M35"><af:mi>G</af:mi></af:math> and its <cf:math xmlns:cf="http://www.w3.org/1998/Math/MathML" id="M36"><cf:mi>l</cf:mi></cf:math> -cliques.