5 lines
40 KiB
XML
5 lines
40 KiB
XML
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-1.74 76.24, 0 63.75 M21.25 85 C9.1 85.39, -0.07 78.33, 0 63.75 M0 63.75 C1.03 46.45, 1.05 29.02, 0 21.25 M0 63.75 C0.06 48.07, -0.24 32.63, 0 21.25 M0 21.25 C-1.02 7.63, 6.99 0.28, 21.25 0 M0 21.25 C-0.04 8.93, 5.78 -1.74, 21.25 0" stroke="#d3d3d3" stroke-width="2" fill="none"></path></g><g transform="translate(314.52604998060156 232.49474585574626) rotate(0 50.508335113525334 37.5)"><text x="50.50833511352539" y="17.619999999999997" font-family="Excalifont, Xiaolai, sans-serif, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="middle" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">let G</text><text x="50.50833511352539" y="42.62" font-family="Excalifont, Xiaolai, sans-serif, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="middle" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">undirected</text><text x="50.50833511352539" y="67.62" font-family="Excalifont, Xiaolai, sans-serif, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="middle" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">graph</text></g><g stroke-linecap="round" transform="translate(160.8871565966864 227.78736487745584) rotate(0 66.24294922157412 42.5)"><path d="M21.25 0 C46.76 2.63, 73.89 2.14, 111.24 0 M21.25 0 C51.85 -0.2, 80.12 0.09, 111.24 0 M111.24 0 C124.24 -1.61, 131.15 5.67, 132.49 21.25 M111.24 0 C125.34 0.41, 133.59 7.2, 132.49 21.25 M132.49 21.25 C133.5 29.17, 134.1 39.86, 132.49 63.75 M132.49 21.25 C131.52 37.58, 131.46 54.85, 132.49 63.75 M132.49 63.75 C132.49 78.37, 126.41 83.69, 111.24 85 M132.49 63.75 C133.67 76.82, 124.01 85.49, 111.24 85 M111.24 85 C77.88 84.16, 48.48 87.68, 21.25 85 M111.24 85 C85.63 84.86, 58.53 84.66, 21.25 85 M21.25 85 C9.07 85.23, -0.05 77.98, 0 63.75 M21.25 85 C6.39 86.49, -1.52 79.26, 0 63.75 M0 63.75 C0 52.42, 0.59 38.99, 0 21.25 M0 63.75 C-0.04 47.96, -0.07 31.29, 0 21.25 M0 21.25 C1.89 8.35, 6.91 0.22, 21.25 0 M0 21.25 C-0.38 5.42, 7.76 1.91, 21.25 0" stroke="#d3d3d3" stroke-width="2" fill="none"></path></g><g transform="translate(189.29677375649294 232.78736487745584) rotate(0 37.83333206176758 37.5)"><text x="37.83333206176758" y="17.619999999999997" font-family="Excalifont, Xiaolai, sans-serif, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="middle" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">let k</text><text x="37.83333206176758" y="42.62" font-family="Excalifont, Xiaolai, sans-serif, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="middle" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">positive</text><text x="37.83333206176758" y="67.62" font-family="Excalifont, Xiaolai, sans-serif, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="middle" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">integer</text></g><g stroke-linecap="round" transform="translate(161.32343486742434 360.90000153359233) rotate(0 133.91047799630036 42.49999999999997)"><path d="M21.25 0 C102.76 -0.92, 180.33 -1.77, 246.57 0 M21.25 0 C77.69 1.88, 134.72 1.27, 246.57 0 M246.57 0 C262.54 0.51, 268.88 7.1, 267.82 21.25 M246.57 0 C262.17 0.72, 269.62 5.31, 267.82 21.25 M267.82 21.25 C265.92 38.21, 268.56 54.01, 267.82 63.75 M267.82 21.25 C268.85 35.7, 267.42 51.68, 267.82 63.75 M267.82 63.75 C267.89 78.94, 259.66 85.74, 246.57 85 M267.82 63.75 C267.91 77.73, 261.97 86.24, 246.57 85 M246.57 85 C164.93 84.99, 84.18 84.91, 21.25 85 M246.57 85 C160.04 82.77, 74.27 84.06, 21.25 85 M21.25 85 C8.12 84.65, 0.38 79.11, 0 63.75 M21.25 85 C8.15 85.6, 1.99 76.61, 0 63.75 M0 63.75 C1.38 51.37, 1.24 42.1, 0 21.25 M0 63.75 C-0.27 47.03, -0.03 30.1, 0 21.25 M0 21.25 C0.36 5.52, 6.27 0.54, 21.25 0 M0 21.25 C0.8 5.84, 5.54 -1.08, 21.25 0" stroke="#d3d3d3" stroke-width="2" fill="none"></path></g><g transform="translate(174.44224874018948 365.90000153359233) rotate(0 120.79166412353516 37.5)"><text x="120.79166412353516" y="17.619999999999997" font-family="Excalifont, Xiaolai, sans-serif, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="middle" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">minimum maximal matching</text><text x="120.79166412353516" y="42.62" font-family="Excalifont, Xiaolai, sans-serif, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="middle" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">on G with at most k</text><text x="120.79166412353516" y="67.62" font-family="Excalifont, Xiaolai, sans-serif, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="middle" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">edges</text></g><g stroke-linecap="round" transform="translate(77.417135795445 499.9689857427033) rotate(0 107.91190115127392 42.38124170627589)"><path d="M21.19 0 C81.37 -2.2, 138.3 -0.1, 194.63 0 M21.19 0 C60.87 0.31, 99.51 0.72, 194.63 0 M194.63 0 C209.7 -1.86, 216.89 6.86, 215.82 21.19 M194.63 0 C206.67 -2.01, 216.04 5.05, 215.82 21.19 M215.82 21.19 C216.6 37.55, 214.48 52.95, 215.82 63.57 M215.82 21.19 C216.03 33.44, 215.1 45.21, 215.82 63.57 M215.82 63.57 C217.07 76.35, 208 85.97, 194.63 84.76 M215.82 63.57 C215.53 79.76, 209.11 86.46, 194.63 84.76 M194.63 84.76 C139.51 87.46, 86.49 84.54, 21.19 84.76 M194.63 84.76 C143.25 83.43, 92.06 83.97, 21.19 84.76 M21.19 84.76 C6.93 86.2, -1.58 79.44, 0 63.57 M21.19 84.76 C8.19 87.06, -0.12 79.32, 0 63.57 M0 63.57 C1.9 51.34, -0.34 43.3, 0 21.19 M0 63.57 C-0.86 49.59, 0.58 36.65, 0 21.19 M0 21.19 C0.32 8.09, 8.1 1.25, 21.19 0 M0 21.19 C-2 9.33, 5.19 0.41, 21.19 0" stroke="#d3d3d3" stroke-width="2" fill="none"></path></g><g transform="translate(119.9457056478908 517.3502274489792) rotate(0 65.38333129882812 25)"><text x="65.38333129882812" y="17.619999999999997" font-family="Excalifont, Xiaolai, sans-serif, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="middle" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">there exists</text><text x="65.38333129882812" y="42.62" font-family="Excalifont, Xiaolai, sans-serif, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="middle" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">matching in G</text></g><g stroke-linecap="round" transform="translate(300.4127171514873 500.26160476441265) rotate(0 112.89792958730652 40.95666215312377)"><path d="M20.48 0 C83.71 0.82, 148.78 -0.8, 205.32 0 M20.48 0 C62.68 0.63, 105.43 0.07, 205.32 0 M205.32 0 C218.87 -0.09, 224.41 6.14, 225.8 20.48 M205.32 0 C218.91 0.81, 228.06 7.42, 225.8 20.48 M225.8 20.48 C225.46 31.18, 224.31 40.51, 225.8 61.43 M225.8 20.48 C225.54 28.72, 225.9 38.26, 225.8 61.43 M225.8 61.43 C226.3 75.37, 220.17 81.88, 205.32 81.91 M225.8 61.43 C226.01 74.61, 220.56 81.1, 205.32 81.91 M205.32 81.91 C164.91 80.9, 124.16 82.26, 20.48 81.91 M205.32 81.91 C152.11 83.63, 97.82 83.87, 20.48 81.91 M20.48 81.91 C8.5 83.61, -1.32 75.67, 0 61.43 M20.48 81.91 C5.52 84.21, 2.02 76.24, 0 61.43 M0 61.43 C0.45 47.97, -1.24 35.92, 0 20.48 M0 61.43 C-0.61 53.07, 0.65 43.79, 0 20.48 M0 20.48 C1.78 8.82, 7.08 -0.55, 20.48 0 M0 20.48 C2.18 6.56, 8.8 -2.21, 20.48 0" stroke="#d3d3d3" stroke-width="2" fill="none"></path></g><g transform="translate(338.0856482646727 516.2182669175364) rotate(0 75.22499847412115 25)"><text x="75.2249984741211" y="17.619999999999997" font-family="Excalifont, Xiaolai, sans-serif, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="middle" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">there exists no</text><text x="75.2249984741211" y="42.62" font-family="Excalifont, Xiaolai, sans-serif, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="middle" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">matching in G</text></g><g stroke-linecap="round"><g transform="translate(300.73439710681225 178.59927291893234) rotate(0 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7.84, -44.06 37.59, -52.63 44.54" stroke="#d3d3d3" stroke-width="2" fill="none"></path></g><g transform="translate(285.6997692717425 178.59927291893234) rotate(0 -26.144248658825802 22.578457447413697)"><path d="M-40.2 22.85 C-42.75 30.39, -49.24 38.29, -52.63 44.54 M-40.2 22.85 C-43.96 31.98, -50.27 39.26, -52.63 44.54" stroke="#d3d3d3" stroke-width="2" fill="none"></path></g><g transform="translate(285.6997692717425 178.59927291893234) rotate(0 -26.144248658825802 22.578457447413697)"><path d="M-29.17 35.92 C-35.75 38.79, -46.21 41.99, -52.63 44.54 M-29.17 35.92 C-37.25 40.01, -47.76 42.3, -52.63 44.54" stroke="#d3d3d3" stroke-width="2" fill="none"></path></g></g><mask></mask><g stroke-linecap="round"><g transform="translate(356.39692521477525 318.49474585574626) rotate(0 -26.35699374932784 18.090991880147186)"><path d="M0.08 -0.62 C-8.54 5.4, -43.38 30.56, -52.07 36.8 M-1.33 1.67 C-10.05 7.25, -44.25 29.37, -52.8 34.88" stroke="#d3d3d3" stroke-width="2" fill="none"></path></g><g transform="translate(356.39692521477525 318.49474585574626) rotate(0 -26.35699374932784 18.090991880147186)"><path d="M-37.7 14.96 C-42.75 21.12, -45.2 26.59, -52.8 34.88 M-37.7 14.96 C-43.05 21.57, -48.36 29.45, -52.8 34.88" stroke="#d3d3d3" stroke-width="2" fill="none"></path></g><g transform="translate(356.39692521477525 318.49474585574626) rotate(0 -26.35699374932784 18.090991880147186)"><path d="M-28.42 29.32 C-36.1 31.56, -41.09 33.09, -52.8 34.88 M-28.42 29.32 C-37.01 31.12, -45.45 34.17, -52.8 34.88" stroke="#d3d3d3" stroke-width="2" fill="none"></path></g></g><mask></mask><g stroke-linecap="round"><g transform="translate(235.63808826001036 318.78736487745584) rotate(0 25.638324928104794 18.027337274322917)"><path d="M-0.51 0.33 C8.09 6.59, 43.35 30.69, 51.79 36.6 M1.42 -0.54 C9.91 5.44, 42.67 28.49, 51.19 34.73" stroke="#d3d3d3" stroke-width="2" fill="none"></path></g><g transform="translate(235.63808826001036 318.78736487745584) rotate(0 25.638324928104794 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transform="translate(308.292598841634 451.9000015335923) rotate(0 45.873403958883046 21.631112475332486)"><path d="M74.29 25.19 C80.46 30.81, 82.05 35.41, 92.04 42.8 M74.29 25.19 C82.02 31.03, 87.66 38.27, 92.04 42.8" stroke="#d3d3d3" stroke-width="2" fill="none"></path></g></g><mask></mask><g stroke-linecap="round" transform="translate(618.9605950826727 233.05625677326515) rotate(0 182.08175006859608 117.5)"><path d="M0 0 C84.34 -1.98, 169.88 -1.73, 364.16 0 M364.16 0 C365.08 58.23, 367.03 115.3, 364.16 235 M364.16 235 C272.14 236.01, 180.94 235.89, 0 235 M0 235 C-2.76 162.13, -1.3 88.42, 0 0" stroke="#d3d3d3" stroke-width="2.5" fill="none" stroke-dasharray="1.5 8"></path></g><g transform="translate(623.9605950826727 238.05625677326515) rotate(0 169.18333435058594 112.5)"><text x="0" y="17.619999999999997" font-family="Excalifont, Xiaolai, sans-serif, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">there are two outcomes of running</text><text x="0" y="42.62" font-family="Excalifont, Xiaolai, sans-serif, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">the algorithm: there is such a</text><text x="0" y="67.62" font-family="Excalifont, Xiaolai, sans-serif, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">matching, or there is no such</text><text x="0" y="92.62" font-family="Excalifont, Xiaolai, sans-serif, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">matching.</text><text x="0" y="117.62" font-family="Excalifont, Xiaolai, sans-serif, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic"></text><text x="0" y="142.62" font-family="Excalifont, Xiaolai, sans-serif, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">we must prove that this decision</text><text x="0" y="167.62" font-family="Excalifont, Xiaolai, sans-serif, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">problem has a polynomial kernel,</text><text x="0" y="192.62" font-family="Excalifont, Xiaolai, sans-serif, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">which would imply that it is fixed-</text><text x="0" y="217.62" font-family="Excalifont, Xiaolai, sans-serif, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">parameter tractable.</text></g><g transform="translate(441.8379069346762 389.42740368659275) rotate(0 45.29999923706055 12.5)"><text x="0" y="17.619999999999997" font-family="Excalifont, Xiaolai, sans-serif, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">black-box</text></g><g stroke-linecap="round" transform="translate(10 10) rotate(0 534.8377248858924 329.4600385297097)"><path d="M0 0 C288.21 0, 576.42 0, 1069.68 0 M0 0 C286.97 0, 573.93 0, 1069.68 0 M1069.68 0 C1069.68 187.95, 1069.68 375.89, 1069.68 658.92 M1069.68 0 C1069.68 178.3, 1069.68 356.59, 1069.68 658.92 M1069.68 658.92 C706.59 658.92, 343.51 658.92, 0 658.92 M1069.68 658.92 C806.7 658.92, 543.73 658.92, 0 658.92 M0 658.92 C0 463.25, 0 267.57, 0 0 M0 658.92 C0 431.15, 0 203.39, 0 0" stroke="#d3d3d3" stroke-width="2" fill="none"></path></g><g stroke-linecap="round" transform="translate(10.00000049771802 677.2105917192328) rotate(0 534.8377248858924 329.46003852970966)"><path d="M0 0 C370.02 0, 740.03 0, 1069.68 0 M0 0 C281.5 0, 563 0, 1069.68 0 M1069.68 0 C1069.68 240.48, 1069.68 480.96, 1069.68 658.92 M1069.68 0 C1069.68 138.26, 1069.68 276.51, 1069.68 658.92 M1069.68 658.92 C739.56 658.92, 409.44 658.92, 0 658.92 M1069.68 658.92 C758.84 658.92, 448 658.92, 0 658.92 M0 658.92 C0 418.9, 0 178.89, 0 0 M0 658.92 C0 510.5, 0 362.08, 0 0" stroke="#d3d3d3" stroke-width="2" fill="none"></path></g><g transform="translate(15.00000049771802 682.2105917192328) rotate(0 505.99999999999994 275)"><text x="0" y="17.5" font-family="Comic Shanns, monospace, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic"></text><text x="0" y="42.5" font-family="Comic Shanns, monospace, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic"> proof:</text><text x="0" y="67.5" font-family="Comic Shanns, monospace, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic"> let M be a maximal matching in G.</text><text x="0" y="92.5" font-family="Comic Shanns, monospace, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic"> (1) if |M| > k, then k is too small</text><text x="0" y="117.5" font-family="Comic Shanns, monospace, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic"> => no such matching found</text><text x="0" y="142.5" font-family="Comic Shanns, monospace, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic"> (2) if |M| <= k, then |M| is within budget</text><text x="0" y="167.5" font-family="Comic Shanns, monospace, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic"> let V(M) denote the vertices covered by M</text><text x="0" y="192.5" font-family="Comic Shanns, monospace, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic"> similar to vertex cover problem</text><text x="0" y="217.5" font-family="Comic Shanns, monospace, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic"> it follows that V(M) <= 2k, since each edge in M covers at most two vertices</text><text x="0" y="242.5" font-family="Comic Shanns, monospace, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic"> let u vertex outside of V(M)</text><text x="0" y="267.5" font-family="Comic Shanns, monospace, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic"> then N(u) can only contain vertices in V(M), since if v in N(u) and v not in V(M)</text><text x="0" y="292.5" font-family="Comic Shanns, monospace, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic"> would mean M is not maximal, thus contradiction.</text><text x="0" y="317.5" font-family="Comic Shanns, monospace, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic"> this gives a reduction rule:</text><text x="0" y="342.5" font-family="Comic Shanns, monospace, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic"> remove all vertices outside of V(M)</text><text x="0" y="367.5" font-family="Comic Shanns, monospace, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic"> then form G[V(M)], the fully connected subgraph induced by V(M)</text><text x="0" y="392.5" font-family="Comic Shanns, monospace, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic"> => return G[V(M)] as kernel</text><text x="0" y="417.5" font-family="Comic Shanns, monospace, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic"> G[V(M)] is a polynomial kernel because</text><text x="0" y="442.5" font-family="Comic Shanns, monospace, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic"> - the number of vertices is at most 2k</text><text x="0" y="467.5" font-family="Comic Shanns, monospace, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic"> - the number of edges is 2k(2k-1)/2 = 2k^2 - k (since fully connected, undirected)</text><text x="0" y="492.5" font-family="Comic Shanns, monospace, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic"> - the kernel size is thus O(k^2), polynomial in k</text><text x="0" y="517.5" font-family="Comic Shanns, monospace, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic"></text><text x="0" y="542.5" font-family="Comic Shanns, monospace, Segoe UI Emoji" font-size="20px" fill="#d3d3d3" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic"> Q.E.D.</text></g></svg> |