Bounds on the Size of the Minimum Dominating Sets of Some Cylindrical Grid Graphs

<table>Modified concatenation of dominating sets of <svg height="15.6875" id="M79" style="vertical-align:-3.25789pt" version="1.1" viewbox="0 0 47.349998 15.6875" width="47.349998" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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</svg> and <svg height="15.6875" id="M80" style="vertical-align:-3.25789pt" version="1.1" viewbox="0 0 47.349998 15.6875" width="47.349998" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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<g transform="matrix(.012,-0,0,-.012,10.875,15.475)"><path d="M153 550l-26 -186q79 31 111 31q90 0 141.5 -51t51.5 -119q0 -93 -89 -166q-85 -69 -173 -71q-32 0 -61.5 11.5t-41.5 23.5q-18 17 -17 34q2 16 22 33q14 9 26 -1q61 -50 124 -50q60 0 93 43.5t33 104.5q0 69 -41.5 110t-121.5 41q-53 0 -102 -20l38 305h286l6 -8
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<g transform="matrix(.017,-0,0,-.017,17.213,11.4)"><path d="M639 0h-586v578h586v-578zM581 50v478h-470v-478h470z" id="x25A1"></path></g><g transform="matrix(.017,-0,0,-.017,28.976,11.4)"><path d="M682 629q-1 -16 -1.5 -72t-2.5 -86l-31 -4q-5 92 -51 129t-139 37q-100 0 -177 -49t-116 -125t-39 -162q0 -122 66 -201t182 -79q83 0 137 42.5t112 128.5l26 -15q-12 -31 -42.5 -88t-45.5 -75q-139 -27 -199 -27q-148 0 -243 81.5t-95 226.5q0 173 129.5 274.5
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</svg> to get dominating set for <svg height="15.725" id="M81" style="vertical-align:-3.32811pt" version="1.1" viewbox="0 0 47.349998 15.725" width="47.349998" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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<g transform="matrix(.017,-0,0,-.017,17.213,11.4)"><path d="M639 0h-586v578h586v-578zM581 50v478h-470v-478h470z" id="x25A1"></path></g><g transform="matrix(.017,-0,0,-.017,28.976,11.4)"><path d="M682 629q-1 -16 -1.5 -72t-2.5 -86l-31 -4q-5 92 -51 129t-139 37q-100 0 -177 -49t-116 -125t-39 -162q0 -122 66 -201t182 -79q83 0 137 42.5t112 128.5l26 -15q-12 -31 -42.5 -88t-45.5 -75q-139 -27 -199 -27q-148 0 -243 81.5t-95 226.5q0 173 129.5 274.5
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</svg>.</table>

International Journal of Combinatorics

fig1

Figure 1

Figure 1: Bounds on the Size of the Minimum Dominating Sets of Some Cylindrical Grid Graphs 