Mathematical Researches

fa احاطه گری هم-رومی در شبکه ها Co-Roman dominating of girds جبر alg علمی پژوهشی بنیادی S <div>فـــرض کنیـــد (G=(V,E  یک گـــراف ســـاده بوده و ، {f:V→{0,1,2 یک تــابــع باشــد کــــه وزن آن به‌صـــورت (w(f تعریف میشود. رأس &lrm; v نسبت به تابع   f محافظت‌شده است هرگاه  0<(f(v یا 0=(v) f و &lrm;v با رأسی با وزن مثبت مجاور باشد. تابع {f:V→{0,1,2  ، یک تابع احاطه‌گر هم-رومی (به اختصار &lrm;CRDF&lrm;) نامیده میشود هرگاه: (1) هر رأس u با وزن صفر حداقل با یک رأس &lrm; v با وزن مثبت مجاور باشد و (2) هر رأس v&lrm;  با وزن مثبت حداقل با یک رأس u با وزن صفر مجاور باشد، بــه طوری‌کــه هر رأس  G نسبت به تابع {f chr(chr(chr('39')39chr('39'))39chr(chr('39')39chr('39'))):V→{0,1,2  ، که با ضابطه ی f chr(chr(chr('39')39chr('39'))39chr(chr('39')39chr('39')))(v)=f(v)-1،  f chr(chr(chr('39')39chr('39'))39chr(chr('39')39chr('39')))(u)=1   ) وf chr(chr(chr('39')39chr('39'))39chr(chr('39')39chr('39')))(x)=f(xبرای سایر رئوس تعریف میشود، محافظت شده باشد. عدد احاطه‌ای هم-رومی گراف G که با نماد (ϫ_cr(G نمایش داده‌ می‌شود، کمترین وزن در بین تمامی توابع احاطه گر هم-رومی گراف  G است.</div> در این مقاله&lrm;&lrm;&lrm;،&lrm; عدد احاطه‌ای هم-رومی شبکه ها را مطالعه کرده و مقدار دقیق این پارامتر را برای شبکه های P2◼Pn و P3◼Pn به دست می‌آوریم&lrm;.   Abstract Let <m:omath><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>G = (V</m:r><m:r>, </m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>E)</m:r></m:omath> be a simple graph with vertex set <m:omath><m:r>V</m:r></m:omath> and let <m:omath><m:r>f</m:r><m:r>:</m:r><m:r>V</m:r><m:r>→</m:r><m:d><m:dpr><m:begchr m:val="{"><m:endchr m:val="}"><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:r>0,1,2</m:r></m:e></m:d><m:r> </m:r></m:omath> be a function of weight <m:omath><m:r>ω</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>f</m:r></m:e></m:d><m:r>=</m:r><m:nary><m:narypr><m:chr m:val="∑"><m:limloc m:val="undOvr"><m:suphide m:val="on"><m:ctrlpr></m:ctrlpr></m:suphide></m:limloc></m:chr></m:narypr><m:sub><m:r>v</m:r><m:r>∈</m:r><m:r>V</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>G</m:r></m:e></m:d></m:sub><m:sup></m:sup><m:e><m:r>f</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>v</m:r></m:e></m:d></m:e></m:nary></m:omath>. A vertex &lrm;<m:omath><m:r>v</m:r></m:omath> is protected with respect to <m:omath><m:r>f</m:r></m:omath>, if <m:omath><m:r>f</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>v</m:r></m:e></m:d><m:r>>0</m:r></m:omath> or <m:omath><m:r>f</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>v</m:r></m:e></m:d><m:r>=0 </m:r></m:omath>and &lrm;<m:omath><m:r>v</m:r></m:omath> is adjacent to a vertex <m:omath><m:r>u</m:r></m:omath> such that <m:omath><m:r>f</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>u</m:r></m:e></m:d><m:r>>0</m:r></m:omath>. The function <m:omath><m:r>f</m:r></m:omath> is a co-Roman dominating function, abbreviated CRDF if: (i) every vertex <m:omath><m:r>u</m:r></m:omath> with <m:omath><m:r>f</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>u</m:r></m:e></m:d><m:r>=0 </m:r></m:omath>is adjacent to a vertex <m:omath><m:r>v</m:r></m:omath> for which <m:omath><m:r>f</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>v</m:r></m:e></m:d><m:r>>0</m:r></m:omath>, and (ii) every vertex <m:omath><m:r>v</m:r></m:omath> with <m:omath><m:r>f</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>v</m:r></m:e></m:d><m:r>>0</m:r></m:omath> has a neighbor <m:omath><m:r>u</m:r></m:omath> for which <m:omath><m:r>f</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>u</m:r></m:e></m:d><m:r>=0</m:r></m:omath>, such that each vertex of <m:omath><m:r>G</m:r></m:omath> is protected with respect to the function <m:omath><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>f</m:r></m:e><m:sup><m:r>'</m:r></m:sup></m:ssup><m:r>:</m:r><m:r>V</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>G</m:r></m:e></m:d><m:r>→</m:r><m:d><m:dpr><m:begchr m:val="{"><m:endchr m:val="}"><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:r>0,1,2</m:r></m:e></m:d></m:omath>, defined by <m:omath><m:r> </m:r><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>f</m:r></m:e><m:sup><m:r>'</m:r></m:sup></m:ssup><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>v</m:r></m:e></m:d><m:r>=</m:r><m:r>f</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>v</m:r></m:e></m:d><m:r>-1</m:r></m:omath>, <m:omath><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>f</m:r></m:e><m:sup><m:r>'</m:r></m:sup></m:ssup><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>u</m:r></m:e></m:d><m:r>=1</m:r></m:omath> and <m:omath><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>f</m:r></m:e><m:sup><m:r>'</m:r></m:sup></m:ssup><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>x</m:r></m:e></m:d><m:r>=</m:r><m:r>f</m:r><m:r>(</m:r><m:r>x</m:r><m:r>)</m:r></m:omath> for <m:omath><m:r>x</m:r><m:r>∈</m:r><m:r>V</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>G</m:r></m:e></m:d><m:r>-</m:r><m:r>{</m:r><m:r>u</m:r><m:r>,</m:r><m:r>v</m:r><m:r>}</m:r></m:omath>. The co-Roman domination number of a graph <m:omath><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>G</m:r></m:omath>, denoted by <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>γ</m:r></m:e><m:sub><m:r>cr</m:r></m:sub></m:ssub><m:r>(</m:r><m:r>G</m:r><m:r>)</m:r></m:omath>, is the minimum weight of a co-Roman dominating function on <m:omath><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>G</m:r></m:omath>. In this paper, we study the co-Roman domination number of grid graphs and we obtain this parameter for <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>P</m:r></m:e><m:sub><m:r>2</m:r></m:sub></m:ssub><m:r>×</m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>P</m:r></m:e><m:sub><m:r>n</m:r></m:sub></m:ssub></m:omath> and <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>P</m:r></m:e><m:sub><m:r>3</m:r></m:sub></m:ssub><m:r>×</m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>P</m:r></m:e><m:sub><m:r>n</m:r></m:sub></m:ssub></m:omath>. Introduction Throughout this paper, <m:omath><m:r>G</m:r></m:omath> is a simple connected graph with vertex set <m:omath><m:r>V</m:r><m:r>=</m:r><m:r>V</m:r><m:r>(</m:r><m:r>G</m:r><m:r>)</m:r></m:omath> and edge set <m:omath><m:r>E</m:r><m:r>=</m:r><m:r>E</m:r><m:r>(</m:r><m:r>G</m:r><m:r>)</m:r></m:omath> of order <m:omath><m:r>n</m:r></m:omath> and size <m:omath><m:r>m</m:r></m:omath>. The Cartesian product, <m:omath><m:r>G</m:r><m:r>×</m:r><m:r>H</m:r></m:omath>, of graphs <m:omath><m:r>G</m:r></m:omath> and <m:omath><m:r>H</m:r></m:omath> is a graph such that the vertex set of <m:omath><m:r> </m:r><m:r>G</m:r><m:r>×</m:r><m:r>H</m:r></m:omath> is the <a href="https://en.wikipedia.org/wiki/Cartesian_product">Cartesian product</a> <m:omath><m:r>V</m:r><m:r>(</m:r><m:r>G</m:r><m:r>) ×</m:r><m:r>V</m:r><m:r>(</m:r><m:r>H</m:r><m:r>)</m:r></m:omath>; and two vertices <m:omath><m:r>(</m:r><m:r>u</m:r><m:r>,</m:r><m:r>u</m:r><m:r>')</m:r></m:omath> and <m:omath><m:r>(</m:r><m:r>v</m:r><m:r>,</m:r><m:r>v</m:r><m:r>')</m:r></m:omath> are adjacent in <m:omath><m:r>G</m:r><m:r>×</m:r><m:r>H</m:r><m:r> </m:r></m:omath>if and only if either <m:omath><m:r>u</m:r><m:r>=</m:r><m:r>v</m:r></m:omath> and <m:omath><m:r>u</m:r><m:r>'</m:r></m:omath> is adjacent to <m:omath><m:r>v</m:r><m:r>'</m:r></m:omath> in <m:omath><m:r>H</m:r></m:omath>, or <m:omath><m:r>u</m:r><m:r>'=</m:r><m:r>v</m:r><m:r>'</m:r></m:omath> and <m:omath><m:r>u</m:r></m:omath> is adjacent to <m:omath><m:r>v</m:r></m:omath> in <m:omath><m:r>G</m:r></m:omath>. The Cartesian product <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>P</m:r></m:e><m:sub><m:r>n</m:r></m:sub></m:ssub><m:r>×</m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>P</m:r></m:e><m:sub><m:r>m</m:r></m:sub></m:ssub></m:omath> is called a grid graph. A set <m:omath><m:r>S</m:r></m:omath> of vertices in a graph <m:omath><m:r>G</m:r></m:omath> is called a dominating set if every vertex in <m:omath><m:r>V</m:r></m:omath> is either an element of <m:omath><m:r>S</m:r></m:omath> or is adjacent to an element of <m:omath><m:r>S</m:r></m:omath>. The domination number of <m:omath><m:r>G</m:r></m:omath>, denoted by <m:omath><m:r>γ</m:r><m:r>(</m:r><m:r>G</m:r><m:r>)</m:r></m:omath>, is the minimum cardinality of a dominating set of <m:omath><m:r>G</m:r></m:omath>. A <m:omath><m:r>γ</m:r><m:r>(</m:r><m:r>G</m:r><m:r>)</m:r></m:omath>-set is a dominating set of <m:omath><m:r>G</m:r></m:omath> of size <m:omath><m:r>γ</m:r><m:r>(</m:r><m:r>G</m:r><m:r>)</m:r></m:omath>. For a subset <m:omath><m:r>S</m:r><m:r>&sube;</m:r><m:r>V</m:r><m:r>(</m:r><m:r>G</m:r><m:r>)</m:r></m:omath> of vertices of a graph <m:omath><m:r>G</m:r></m:omath> and a function <m:omath><m:r>f</m:r><m:r>:</m:r><m:r>V</m:r><m:r>(</m:r><m:r>G</m:r><m:r><m:rpr><m:scr m:val="double-struck"><m:sty m:val="i"></m:sty></m:scr></m:rpr>)→R</m:r></m:omath>, we define <m:omath><m:r>f</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>S</m:r></m:e></m:d><m:r>=</m:r><m:nary><m:narypr><m:chr m:val="∑"><m:limloc m:val="undOvr"><m:suphide m:val="on"><m:ctrlpr></m:ctrlpr></m:suphide></m:limloc></m:chr></m:narypr><m:sub><m:r>v</m:r><m:r>∈</m:r><m:r>S</m:r></m:sub><m:sup></m:sup><m:e><m:r>f</m:r><m:r>(</m:r><m:r>S</m:r><m:r>)</m:r></m:e></m:nary></m:omath>. The weight of <m:omath><m:r>f</m:r></m:omath> is <m:omath><m:r>ω</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>f</m:r></m:e></m:d><m:r>=</m:r><m:r>f</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>V</m:r><m:r>(</m:r><m:r>G</m:r><m:r>)</m:r></m:e></m:d><m:r>=</m:r><m:nary><m:narypr><m:chr m:val="∑"><m:limloc m:val="undOvr"><m:suphide m:val="on"><m:ctrlpr></m:ctrlpr></m:suphide></m:limloc></m:chr></m:narypr><m:sub><m:r>v</m:r><m:r>∈</m:r><m:r>V</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>G</m:r></m:e></m:d></m:sub><m:sup></m:sup><m:e><m:r>f</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>v</m:r></m:e></m:d></m:e></m:nary></m:omath>. A Roman dominating function on a graph <m:omath><m:r>G</m:r></m:omath>, abbreviated RD-function, is a function <m:omath><m:r>f</m:r><m:r>:</m:r><m:r>V</m:r><m:r>→</m:r><m:d><m:dpr><m:begchr m:val="{"><m:endchr m:val="}"><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:r>0,1,2</m:r></m:e></m:d></m:omath> satisfying the condition that every vertex <m:omath><m:r>u</m:r></m:omath> for which <m:omath><m:r>f</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>u</m:r></m:e></m:d><m:r>=0</m:r></m:omath> is adjacent to at least one vertex <m:omath><m:r>v</m:r></m:omath> for which <m:omath><m:r>f</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>v</m:r></m:e></m:d><m:r>=2</m:r></m:omath>. The minimum weight of an RD-function on <m:omath><m:r>G</m:r></m:omath> is called the Roman domination number of <m:omath><m:r>G</m:r></m:omath> and is denoted by <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>γ</m:r></m:e><m:sub><m:r>R</m:r></m:sub></m:ssub><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>G</m:r></m:e></m:d></m:omath>. An RD-function with minimum weight <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>γ</m:r></m:e><m:sub><m:r>R</m:r></m:sub></m:ssub><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>G</m:r></m:e></m:d></m:omath> in <m:omath><m:r>G</m:r></m:omath> is called a <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>γ</m:r></m:e><m:sub><m:r>R</m:r></m:sub></m:ssub><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>G</m:r></m:e></m:d></m:omath>-function of <m:omath><m:r>G</m:r></m:omath>. The Roman domination was introduced by Cockayne et al. in [5]. Let <m:omath><m:r>f</m:r><m:r>:</m:r><m:r>V</m:r><m:r>→</m:r><m:d><m:dpr><m:begchr m:val="{"><m:endchr m:val="}"><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:r>0,1,2</m:r></m:e></m:d><m:r> </m:r></m:omath> be a function. A vertex &lrm;<m:omath><m:r>v</m:r></m:omath> is protected with respect to <m:omath><m:r>f</m:r></m:omath>, if <m:omath><m:r>f</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>v</m:r></m:e></m:d><m:r>>0</m:r></m:omath> or <m:omath><m:r>f</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>v</m:r></m:e></m:d><m:r>=0 </m:r></m:omath>and &lrm;<m:omath><m:r>v</m:r></m:omath> is adjacent to a vertex <m:omath><m:r>u</m:r></m:omath> such that <m:omath><m:r>f</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>u</m:r></m:e></m:d><m:r>>0</m:r></m:omath>. The function <m:omath><m:r>f</m:r></m:omath> is a co-Roman dominating function, abbreviated CRDF if: (i) every vertex <m:omath><m:r>u</m:r></m:omath> with <m:omath><m:r>f</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>u</m:r></m:e></m:d><m:r>=0 </m:r></m:omath>is adjacent to a vertex <m:omath><m:r>v</m:r></m:omath> for which <m:omath><m:r>f</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>v</m:r></m:e></m:d><m:r>>0</m:r></m:omath>, and (ii) every vertex <m:omath><m:r>v</m:r></m:omath> with <m:omath><m:r>f</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>v</m:r></m:e></m:d><m:r>>0</m:r></m:omath> has a neighbor <m:omath><m:r>u</m:r></m:omath> for which <m:omath><m:r>f</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>u</m:r></m:e></m:d><m:r>=0  </m:r></m:omath>such that each vertex of <m:omath><m:r>G</m:r></m:omath> is protected with respect to the function <m:omath><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>f</m:r></m:e><m:sup><m:r>'</m:r></m:sup></m:ssup><m:r>:</m:r><m:r>V</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>G</m:r></m:e></m:d><m:r>→</m:r><m:d><m:dpr><m:begchr m:val="{"><m:endchr m:val="}"><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:r>0,1,2</m:r></m:e></m:d></m:omath>, defined by <m:omath><m:r> </m:r><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>f</m:r></m:e><m:sup><m:r>'</m:r></m:sup></m:ssup><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>v</m:r></m:e></m:d><m:r>=</m:r><m:r>f</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>v</m:r></m:e></m:d><m:r>-1</m:r></m:omath>, <m:omath><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>f</m:r></m:e><m:sup><m:r>'</m:r></m:sup></m:ssup><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>u</m:r></m:e></m:d><m:r>=1</m:r></m:omath> and <m:omath><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>f</m:r></m:e><m:sup><m:r>'</m:r></m:sup></m:ssup><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>x</m:r></m:e></m:d><m:r>=</m:r><m:r>f</m:r><m:r>(</m:r><m:r>x</m:r><m:r>)</m:r></m:omath> for <m:omath><m:r>x</m:r><m:r>∈</m:r><m:r>V</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>G</m:r></m:e></m:d><m:r>-</m:r><m:r>{</m:r><m:r>u</m:r><m:r>,</m:r><m:r>v</m:r><m:r>}</m:r></m:omath>. The co-Roman domination number of a graph <m:omath><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>G</m:r></m:omath>, denoted by <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>γ</m:r></m:e><m:sub><m:r>cr</m:r></m:sub></m:ssub><m:r>(</m:r><m:r>G</m:r><m:r>)</m:r></m:omath>, is the minimum weight of a co-Roman dominating function on <m:omath><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>G</m:r></m:omath>. The concept of co-Roman domination was introduced by Arumugam and et. al [2] and was studied by Shao and et. al [9]. In this paper, we study the co-Roman domination number of grid graphs and we obtain this parameter for <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>P</m:r></m:e><m:sub><m:r>2</m:r></m:sub></m:ssub><m:r>×</m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>P</m:r></m:e><m:sub><m:r>n</m:r></m:sub></m:ssub></m:omath> and <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>P</m:r></m:e><m:sub><m:r>3</m:r></m:sub></m:ssub><m:r>×</m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>P</m:r></m:e><m:sub><m:r>n</m:r></m:sub></m:ssub></m:omath>. For a more thorough treatment of domination parameters and for terminology not presented here see [6], [7] and [10]. The following results are useful in this paper. Proposition 1. [2] For a cycle <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>C</m:r></m:e><m:sub><m:r>n</m:r></m:sub></m:ssub></m:omath>, <m:omath><m:r>n</m:r><m:r>≥4</m:r></m:omath>, and for a path <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>P</m:r></m:e><m:sub><m:r>n</m:r></m:sub></m:ssub></m:omath>, <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>γ</m:r></m:e><m:sub><m:r>cr</m:r></m:sub></m:ssub><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>C</m:r></m:e><m:sub><m:r>n</m:r></m:sub></m:ssub></m:e></m:d><m:r>=</m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>γ</m:r></m:e><m:sub><m:r>cr</m:r></m:sub></m:ssub><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>P</m:r></m:e><m:sub><m:r>n</m:r></m:sub></m:ssub></m:e></m:d><m:r>=</m:r><m:d><m:dpr><m:begchr m:val="⌈"><m:endchr m:val="⌉"><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:f><m:fpr><m:ctrlpr></m:ctrlpr></m:fpr><m:num><m:r>2 </m:r><m:r>n</m:r></m:num><m:den><m:r>5</m:r></m:den></m:f></m:e></m:d></m:omath>. Proposition 2. [2]. For a graph <m:omath><m:r>G</m:r></m:omath>, <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>γ</m:r></m:e><m:sub><m:r>cr</m:r></m:sub></m:ssub><m:r>≤</m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>γ</m:r></m:e><m:sub><m:r>R</m:r></m:sub></m:ssub><m:r>≤2 </m:r><m:r>γ</m:r></m:omath>. Proposition 3. [2] For any tree <m:omath><m:r>T</m:r><m:r> </m:r></m:omath>of order <m:omath><m:r>n</m:r></m:omath>, &lrm; <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>γ</m:r></m:e><m:sub><m:r>cr</m:r></m:sub></m:ssub><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>T</m:r></m:e></m:d><m:r>≤</m:r><m:f><m:fpr><m:ctrlpr></m:ctrlpr></m:fpr><m:num><m:r>2 </m:r><m:r>n</m:r></m:num><m:den><m:r>3</m:r></m:den></m:f></m:omath>. Main Results Theorem 1.     For <m:omath><m:r>n</m:r><m:r>≥2</m:r></m:omath>, <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>γ</m:r></m:e><m:sub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>cr</m:r></m:sub></m:ssub><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>P</m:r></m:e><m:sub><m:r>2</m:r></m:sub></m:ssub><m:r>× </m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>P</m:r></m:e><m:sub><m:r>n</m:r></m:sub></m:ssub></m:e></m:d><m:r>=</m:r><m:d><m:dpr><m:begchr m:val="{"><m:endchr m:val=""><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:m><m:mpr><m:mcs><m:mc><m:mcpr><m:count m:val="2"><m:mcjc m:val="center"></m:mcjc></m:count></m:mcpr></m:mc></m:mcs><m:ctrlpr></m:ctrlpr></m:mpr><m:mr><m:e><m:d><m:dpr><m:begchr m:val="⌈"><m:endchr m:val="⌉"><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:f><m:fpr><m:ctrlpr></m:ctrlpr></m:fpr><m:num><m:r>2 </m:r><m:r>n</m:r></m:num><m:den><m:r>3</m:r></m:den></m:f></m:e></m:d><m:r>+1</m:r></m:e><m:e><m:r>n</m:r><m:r>&equiv;0,1 (</m:r><m:r>mod</m:r><m:r> 3)</m:r></m:e></m:mr><m:mr><m:e><m:d><m:dpr><m:begchr m:val="⌈"><m:endchr m:val="⌉"><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:f><m:fpr><m:ctrlpr></m:ctrlpr></m:fpr><m:num><m:r>2 </m:r><m:r>n</m:r></m:num><m:den><m:r>3</m:r></m:den></m:f></m:e></m:d></m:e><m:e><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>otherwise</m:r></m:e></m:mr></m:m></m:e></m:d></m:omath>. Theorem 2.     For <m:omath><m:r>n</m:r><m:r>≥2</m:r></m:omath>, &lrm;<m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>γ</m:r></m:e><m:sub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>cr</m:r></m:sub></m:ssub><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>P</m:r></m:e><m:sub><m:r>3</m:r></m:sub></m:ssub><m:r>× </m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>P</m:r></m:e><m:sub><m:r>n</m:r></m:sub></m:ssub></m:e></m:d><m:r>=</m:r><m:d><m:dpr><m:begchr m:val="{"><m:endchr m:val=""><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:m><m:mpr><m:mcs><m:mc><m:mcpr><m:count m:val="2"><m:mcjc m:val="center"></m:mcjc></m:count></m:mcpr></m:mc></m:mcs><m:ctrlpr></m:ctrlpr></m:mpr><m:mr><m:e><m:r>n</m:r></m:e><m:e><m:r>n</m:r><m:r>&equiv;1 (</m:r><m:r>mod</m:r><m:r> 3)</m:r></m:e></m:mr><m:mr><m:e><m:r>n</m:r><m:r>+1</m:r></m:e><m:e><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>otherwise</m:r></m:e></m:mr></m:m></m:e></m:d></m:omath>. 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Prentice Hall Inc., Upper Saddle River, Ǌ, (2001) 2nd ed.   تابع احاطه‌گر رومی- تابع احاطه‌گر هم-رومی, شبکه, عدد احاطه‌ای رومی, عدد احاطه‌ای هم-رومی. Roman dominating function, co-Roman dominating function, grid, Roman domination number, co-Roman domination number. 80 90 http://mmr.khu.ac.ir/browse.php?a_code=A-10-806-2&slc_lang=fa&sid=1 Rana Khoeilr رعنا خوئیلر khoeilar@azaruniv.ac.ir 10031947532846005780 10031947532846005780 Yes Azararbaijan Shahid Madani University دانشگاه شهید مدنی آذربایجان marzieh soroudi مرضیه سرودی m.soroudi@azaruniv.ac.ir 10031947532846005781 10031947532846005781 No Azararbaijan Shahid Madani University دانشگاه شهید مدنی آذربایجان Maryam Atapour مریم عطاپور m.atapour@ubonab.ac.ir 10031947532846005782 10031947532846005782 No university of Bonab دانشگاه بناب