From 940310cfd1cba340b2e3509dafccb980e101fb17 Mon Sep 17 00:00:00 2001 From: Andrew Guschin Date: Wed, 30 Oct 2024 21:09:14 +0400 Subject: fix(nir): update tables with correct data --- nir/nir.tex | 378 ++++++++++++++++++++++++++++-------------------------------- 1 file changed, 176 insertions(+), 202 deletions(-) (limited to 'nir/nir.tex') diff --git a/nir/nir.tex b/nir/nir.tex index 2a5a009..cd3e5bb 100644 --- a/nir/nir.tex +++ b/nir/nir.tex @@ -5,7 +5,7 @@ \course{5} \group{531} \author{Гущина Андрея Юрьевича} -\satitle{д.ф.-м.н., доцент} +\satitle{д.ф.-м.н., профессор} \saname{М. Б. Абросимов} \date{2024} @@ -349,8 +349,8 @@ $fg(G)$. В первой колонке указаны возможные зна \begin{tabular}{|c|c|c|c|c|c|c|} \hline $i(G)$ \textbackslash{} $fg(G)$ & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline - 1 & 8 & 1 & 1 & 0 & 0 & 0 \\ \hline - 2 & 16 & 1 & 0 & 0 & 0 & 0 \\ \hline + 1 & 9 & 1 & 1 & 0 & 0 & 0 \\ \hline + 2 & 15 & 1 & 0 & 0 & 0 & 0 \\ \hline 3 & 5 & 0 & 0 & 0 & 0 & 0 \\ \hline 4 & 1 & 0 & 0 & 0 & 0 & 0 \\ \hline 5 & 1 & 0 & 0 & 0 & 0 & 0 \\ \hline @@ -364,10 +364,10 @@ $fg(G)$. В первой колонке указаны возможные зна \begin{tabular}{|c|c|c|c|c|c|c|c|} \hline $i(G)$ \textbackslash{} $fg(G)$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline - 1 & 25 & 5 & 3 & 0 & 0 & 0 & 0 \\ \hline - 2 & 66 & 11 & 7 & 2 & 0 & 0 & 0 \\ \hline - 3 & 25 & 2 & 2 & 1 & 0 & 0 & 0 \\ \hline - 4 & 5 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline + 1 & 26 & 5 & 3 & 0 & 0 & 0 & 0 \\ \hline + 2 & 65 & 12 & 7 & 2 & 0 & 0 & 0 \\ \hline + 3 & 22 & 1 & 2 & 1 & 0 & 0 & 0 \\ \hline + 4 & 8 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline 5 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline 6 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \end{tabular} @@ -380,11 +380,11 @@ $fg(G)$. В первой колонке указаны возможные зна \begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline $i(G)$ \textbackslash{} $fg(G)$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline - 1 & 59 & 20 & 8 & 2 & 0 & 0 & 0 & 0 \\ \hline - 2 & 373 & 129 & 62 & 10 & 0 & 0 & 0 & 0 \\ \hline - 3 & 210 & 48 & 26 & 1 & 0 & 0 & 0 & 0 \\ \hline - 4 & 73 & 3 & 7 & 0 & 0 & 0 & 0 & 0 \\ \hline - 5 & 11 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline + 1 & 95 & 34 & 16 & 6 & 0 & 0 & 0 & 0 \\ \hline + 2 & 437 & 145 & 72 & 6 & 0 & 0 & 0 & 0 \\ \hline + 3 & 160 & 21 & 13 & 1 & 0 & 0 & 0 & 0 \\ \hline + 4 & 28 & 0 & 2 & 0 & 0 & 0 & 0 & 0 \\ \hline + 5 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline 6 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline 7 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \end{tabular} @@ -397,12 +397,12 @@ $fg(G)$. В первой колонке указаны возможные зна \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|} \hline $i(G)$ \textbackslash{} $fg(G)$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline - 1 & 485 & 254 & 93 & 27 & 0 & 0 & 0 & 0 & 0 \\ \hline - 2 & 4569 & 2059 & 897 & 133 & 1 & 0 & 0 & 0 & 0 \\ \hline - 3 & 1739 & 555 & 200 & 15 & 2 & 0 & 0 & 0 & 0 \\ \hline - 4 & 801 & 206 & 78 & 2 & 1 & 0 & 0 & 0 & 0 \\ \hline - 5 & 183 & 14 & 10 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline - 6 & 19 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline + 1 & 565 & 299 & 114 & 29 & 0 & 0 & 0 & 0 & 0 \\ \hline + 2 & 5171 & 2240 & 957 & 134 & 3 & 0 & 0 & 0 & 0 \\ \hline + 3 & 1749 & 529 & 198 & 12 & 1 & 0 & 0 & 0 & 0 \\ \hline + 4 & 261 & 21 & 9 & 2 & 0 & 0 & 0 & 0 & 0 \\ \hline + 5 & 44 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline + 6 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline 7 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline 8 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \end{tabular} @@ -415,13 +415,13 @@ $fg(G)$. В первой колонке указаны возможные зна \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|} \hline $i(G)$ \textbackslash{} $fg(G)$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline - 1 & 6110 & 4128 & 1279 & 379 & 23 & 0 & 0 & 0 & 0 & 0 \\ \hline - 2 & 108690 & 63700 & 20121 & 3381 & 150 & 0 & 0 & 0 & 0 & 0 \\ \hline - 3 & 34947 & 18494 & 5062 & 490 & 17 & 0 & 0 & 0 & 0 & 0 \\ \hline - 4 & 4074 & 1087 & 293 & 22 & 3 & 0 & 0 & 0 & 0 & 0 \\ \hline - 5 & 1307 & 360 & 84 & 3 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline - 6 & 395 & 34 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline - 7 & 25 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline + 1 & 6198 & 4196 & 1310 & 379 & 23 & 0 & 0 & 0 & 0 & 0 \\ \hline + 2 & 109660 & 64137 & 20190 & 3389 & 150 & 0 & 0 & 0 & 0 & 0 \\ \hline + 3 & 35432 & 18557 & 5054 & 488 & 19 & 0 & 0 & 0 & 0 & 0 \\ \hline + 4 & 3744 & 840 & 268 & 19 & 1 & 0 & 0 & 0 & 0 & 0 \\ \hline + 5 & 437 & 72 & 24 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline + 6 & 70 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline + 7 & 7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline 8 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline 9 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \end{tabular} @@ -434,15 +434,15 @@ $fg(G)$. В первой колонке указаны возможные зна \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline $i(G)$ \textbackslash{} $fg(G)$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline - 1 & 126042 & 105165 & 25873 & 8402 & 973 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline - 2 & 4305543 & 3337594 & 787147 & 164044 & 19062 & 11 & 0 & 0 & 0 & 0 & 0 \\ \hline - 3 & 1434656 & 1137131 & 237609 & 39140 & 3038 & 3 & 0 & 0 & 0 & 0 & 0 \\ \hline - 4 & 141618 & 72118 & 15646 & 1907 & 289 & 1 & 0 & 0 & 0 & 0 & 0 \\ \hline - 5 & 23836 & 12234 & 3025 & 258 & 20 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline - 6 & 2065 & 479 & 130 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline - 7 & 91 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline + 1 & 128633 & 107399 & 26348 & 8563 & 996 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline + 2 & 4360884 & 3381726 & 796931 & 165666 & 19278 & 11 & 0 & 0 & 0 & 0 & 0 \\ \hline + 3 & 1434128 & 1135938 & 237113 & 38740 & 2989 & 3 & 0 & 0 & 0 & 0 & 0 \\ \hline + 4 & 100512 & 36441 & 8179 & 714 & 113 & 1 & 0 & 0 & 0 & 0 & 0 \\ \hline + 5 & 8700 & 3023 & 801 & 71 & 6 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline + 6 & 947 & 198 & 58 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline + 7 & 46 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline 8 & 7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline - 9 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline + 9 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline 10 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \end{tabular} \label{tbl:gr10} @@ -486,7 +486,7 @@ $G_0 = \set{0, 1, 2}$, $G_1 = \set{3, 4, 5}$). \begin{figure}[h] \centering - \begin{tikzpicture}[scale=4] + \begin{tikzpicture}[scale=3] \draw (1.0, 0.0) node[line width=1.5, draw=black, circle] (0){0} (0.5, 0.866) node[line width=1.5, draw=black, circle] (1){1} @@ -510,16 +510,9 @@ $G_0 = \set{0, 1, 2}$, $G_1 = \set{3, 4, 5}$). \caption{Визуализация графа <>} \label{fig:fg_even} \end{figure} - -Если $n$ --- нечётное, то $fg(G) = \lfloor \frac{n}{2} \rfloor$ только в том -случае, если существует два минимальных геодезических -множества мощности $(\lfloor \frac{n}{2} \rfloor + 1)$, пересекающихся в одной -вершине (например, рис. -\ref{fig:fg_odd}, где $G_0 = \set{0, 1, 2, 3}$, $G_1 = \set{0, 4, 5, 6}$). - \begin{figure}[h] \centering - \begin{tikzpicture}[scale=4] + \begin{tikzpicture}[scale=3] \draw (1.0, 0.0) node[line width=1.5, draw=black, circle] (0){0} (0.623, 0.782) node[line width=1.5, draw=black, circle] (1){1} @@ -548,19 +541,21 @@ $G_0 = \set{0, 1, 2}$, $G_1 = \set{3, 4, 5}$). \label{fig:fg_odd} \end{figure} +Если $n$ --- нечётное, то $fg(G) = \lfloor \frac{n}{2} \rfloor$ только в том +случае, если существует два минимальных геодезических +множества мощности $(\lfloor \frac{n}{2} \rfloor + 1)$, пересекающихся в одной +вершине (например, рис. +\ref{fig:fg_odd}, где $G_0 = \set{0, 1, 2, 3}$, $G_1 = \set{0, 4, 5, 6}$). + + Помимо этого, можно увидеть, что при $fg(G) = 0$ значения $i(G)$ варьируются от 1 до $n$. Очевидно, что у всех несвязных графов $fg(G) = 0$, поэтому они находятся только в этих колонках. -Но при $fg(G) > 0$ значение $i(G) < (n - 2)$ (при $n > 4$). -Необходимо отметить, что $i(G) = n$ может быть только в случае, если граф -пустой. Но в пустом графе не может быть $fg(G) > 0$. Поэтому $i(G) = n$ и -$fg(G) > 0$ не может быть ни у одного графа. $i(G) = 1$ при этом может быть -только у полного графа. Но у полного графа минимальными геодезическими -множествами являются любые две пары вершин. То есть ни в одной из таких пар нет -уникальной вершины, находящейся только в этой паре. Таким образом, $fg(G) = 0$. -Для всех остальных графов не может быть справедливо одновременно $i(G) = 1$ и -$fg(G) > 0$. +Но при $fg(G) > 0$ значение $i(G) < (n - 2)$ (при $n > 4$). Необходимо +отметить, что $i(G) = n$ может быть только в случае, если граф пустой. Но в +пустом графе не может быть $fg(G) > 0$. Поэтому $i(G) = n$ и $fg(G) > 0$ не +может быть ни у одного графа. Рассмотрим случай с $i(G) = n - 2$. Получаем, что существует $n - 2$ вершины (обозначим множеством $V_i$), не смежные между собой и каждая из них смежна @@ -612,9 +607,8 @@ $fg(G)$. \draw[line width=1.5] (1) to (3); \end{scope} \end{tikzpicture} - \caption{Визуализация графа <>, $i(G) = 2, fg(G) = 2$} + \caption{Визуализация графа <>, $i(G) = 2$, $fg(G) = 2$} \end{figure} - \begin{figure}[h] \centering \begin{tikzpicture}[scale=4] @@ -635,9 +629,8 @@ $fg(G)$. \draw[line width=1.5] (3) to (4); \end{scope} \end{tikzpicture} - \caption{Визуализация графа <>, $i(G) = 2, fg(G) = 2$} + \caption{Визуализация графа <>, $i(G) = 1$, $fg(G) = 2$} \end{figure} - \begin{figure}[h] \centering \begin{tikzpicture}[scale=4] @@ -656,9 +649,8 @@ $fg(G)$. \draw[line width=1.5] (4) to (5); \end{scope} \end{tikzpicture} - \caption{Визуализация графа <>, $i(G) = 4, fg(G) = 0$} + \caption{Визуализация графа <>, $i(G) = 2$, $fg(G) = 0$} \end{figure} - \begin{figure}[h] \centering \begin{tikzpicture}[scale=4] @@ -678,9 +670,8 @@ $fg(G)$. \draw[line width=1.5] (3) to (5); \end{scope} \end{tikzpicture} - \caption{Визуализация графа <>, $i(G) = 3, fg(G) = 1$} + \caption{Визуализация графа <>, $i(G) = 2$, $fg(G) = 1$} \end{figure} - \begin{figure}[h] \centering \begin{tikzpicture}[scale=4] @@ -705,9 +696,8 @@ $fg(G)$. \draw[line width=1.5] (5) to (6); \end{scope} \end{tikzpicture} - \caption{Визуализация графа <>, $i(G) = 4, fg(G) = 2$} + \caption{Визуализация графа <>, $i(G) = 4$, $fg(G) = 2$} \end{figure} - \begin{figure}[h] \centering \begin{tikzpicture}[scale=4] @@ -743,9 +733,8 @@ $fg(G)$. \draw[line width=1.5] (5) to (6); \end{scope} \end{tikzpicture} - \caption{Визуализация графа <>, $i(G) = 1, fg(G) = 0$} + \caption{Визуализация графа <>, $i(G) = 1$, $fg(G) = 0$} \end{figure} - \begin{figure}[h] \centering \begin{tikzpicture}[scale=4] @@ -759,21 +748,13 @@ $fg(G)$. (0.0, -1.0) node[line width=1.5, draw=black, circle] (6){6} (0.707, -0.707) node[line width=1.5, draw=black, circle] (7){7}; \begin{scope}[-] - \draw[line width=1.5] (0) to (4); - \draw[line width=1.5] (0) to (5); - \draw[line width=1.5] (1) to (4); + \draw[line width=1.5] (0) to (7); \draw[line width=1.5] (1) to (7); - \draw[line width=1.5] (2) to (5); - \draw[line width=1.5] (2) to (6); - \draw[line width=1.5] (3) to (6); - \draw[line width=1.5] (3) to (7); - \draw[line width=1.5] (4) to (6); - \draw[line width=1.5] (5) to (7); + \draw[line width=1.5] (2) to (7); \end{scope} \end{tikzpicture} - \caption{Визуализация графа <>, $i(G) = 4, fg(G) = 4$} + \caption{Визуализация графа <>, $i(G) = 5$, $fg(G) = 0$} \end{figure} - \begin{figure}[h] \centering \begin{tikzpicture}[scale=4] @@ -787,24 +768,17 @@ $fg(G)$. (0.0, -1.0) node[line width=1.5, draw=black, circle] (6){6} (0.707, -0.707) node[line width=1.5, draw=black, circle] (7){7}; \begin{scope}[-] - \draw[line width=1.5] (0) to (3); \draw[line width=1.5] (0) to (5); \draw[line width=1.5] (0) to (6); - \draw[line width=1.5] (1) to (4); - \draw[line width=1.5] (1) to (5); \draw[line width=1.5] (1) to (6); - \draw[line width=1.5] (2) to (5); + \draw[line width=1.5] (1) to (7); \draw[line width=1.5] (2) to (6); - \draw[line width=1.5] (2) to (7); - \draw[line width=1.5] (3) to (5); \draw[line width=1.5] (3) to (7); - \draw[line width=1.5] (4) to (6); \draw[line width=1.5] (4) to (7); \draw[line width=1.5] (5) to (7); - \draw[line width=1.5] (6) to (7); \end{scope} \end{tikzpicture} - \caption{Визуализация графа <>, $i(G) = 3, fg(G) = 4$} + \caption{Визуализация графа <>, $i(G) = 2$, $fg(G) = 1$} \end{figure} \begin{figure}[h] @@ -820,22 +794,19 @@ $fg(G)$. (0.0, -1.0) node[line width=1.5, draw=black, circle] (6){6} (0.707, -0.707) node[line width=1.5, draw=black, circle] (7){7}; \begin{scope}[-] - \draw[line width=1.5] (0) to (3); - \draw[line width=1.5] (0) to (6); - \draw[line width=1.5] (0) to (7); + \draw[line width=1.5] (0) to (4); + \draw[line width=1.5] (0) to (5); \draw[line width=1.5] (1) to (4); - \draw[line width=1.5] (1) to (6); \draw[line width=1.5] (1) to (7); \draw[line width=1.5] (2) to (5); \draw[line width=1.5] (2) to (6); - \draw[line width=1.5] (2) to (7); \draw[line width=1.5] (3) to (6); \draw[line width=1.5] (3) to (7); \draw[line width=1.5] (4) to (6); \draw[line width=1.5] (5) to (7); \end{scope} \end{tikzpicture} - \caption{Визуализация графа <>, $i(G) = 3, fg(G) = 2$} + \caption{Визуализация графа <>, $i(G) = 3$, $fg(G) = 4$} \end{figure} \begin{figure}[h] @@ -851,14 +822,23 @@ $fg(G)$. (0.0, -1.0) node[line width=1.5, draw=black, circle] (6){6} (0.707, -0.707) node[line width=1.5, draw=black, circle] (7){7}; \begin{scope}[-] + \draw[line width=1.5] (0) to (3); + \draw[line width=1.5] (0) to (6); \draw[line width=1.5] (0) to (7); + \draw[line width=1.5] (1) to (4); + \draw[line width=1.5] (1) to (6); \draw[line width=1.5] (1) to (7); + \draw[line width=1.5] (2) to (5); + \draw[line width=1.5] (2) to (6); \draw[line width=1.5] (2) to (7); + \draw[line width=1.5] (3) to (6); + \draw[line width=1.5] (3) to (7); + \draw[line width=1.5] (4) to (6); + \draw[line width=1.5] (5) to (7); \end{scope} \end{tikzpicture} - \caption{Визуализация графа <>, $i(G) = 7, fg(G) = 0$} + \caption{Визуализация графа <>, $i(G) = 2$, $fg(G) = 2$} \end{figure} - \begin{figure}[h] \centering \begin{tikzpicture}[scale=4] @@ -872,19 +852,25 @@ $fg(G)$. (0.0, -1.0) node[line width=1.5, draw=black, circle] (6){6} (0.707, -0.707) node[line width=1.5, draw=black, circle] (7){7}; \begin{scope}[-] + \draw[line width=1.5] (0) to (3); \draw[line width=1.5] (0) to (5); \draw[line width=1.5] (0) to (6); + \draw[line width=1.5] (1) to (4); + \draw[line width=1.5] (1) to (5); \draw[line width=1.5] (1) to (6); - \draw[line width=1.5] (1) to (7); + \draw[line width=1.5] (2) to (5); \draw[line width=1.5] (2) to (6); + \draw[line width=1.5] (2) to (7); + \draw[line width=1.5] (3) to (5); \draw[line width=1.5] (3) to (7); + \draw[line width=1.5] (4) to (6); \draw[line width=1.5] (4) to (7); \draw[line width=1.5] (5) to (7); + \draw[line width=1.5] (6) to (7); \end{scope} \end{tikzpicture} - \caption{Визуализация графа <>, $i(G) = 5, fg(G) = 1$} + \caption{Визуализация графа <>, $i(G) = 2$, $fg(G) = 4$} \end{figure} - \begin{figure}[h] \centering \begin{tikzpicture}[scale=4] @@ -899,38 +885,25 @@ $fg(G)$. (0.174, -0.985) node[line width=1.5, draw=black, circle] (7){7} (0.766, -0.643) node[line width=1.5, draw=black, circle] (8){8}; \begin{scope}[-] - \draw[line width=1.5] (0) to (2); - \draw[line width=1.5] (0) to (4); - \draw[line width=1.5] (0) to (5); \draw[line width=1.5] (0) to (6); \draw[line width=1.5] (0) to (7); \draw[line width=1.5] (0) to (8); - \draw[line width=1.5] (1) to (3); - \draw[line width=1.5] (1) to (4); - \draw[line width=1.5] (1) to (6); \draw[line width=1.5] (1) to (7); \draw[line width=1.5] (1) to (8); - \draw[line width=1.5] (2) to (4); - \draw[line width=1.5] (2) to (5); - \draw[line width=1.5] (2) to (6); \draw[line width=1.5] (2) to (7); \draw[line width=1.5] (2) to (8); - \draw[line width=1.5] (3) to (5); - \draw[line width=1.5] (3) to (6); \draw[line width=1.5] (3) to (7); - \draw[line width=1.5] (4) to (5); - \draw[line width=1.5] (4) to (6); + \draw[line width=1.5] (3) to (8); + \draw[line width=1.5] (4) to (7); \draw[line width=1.5] (4) to (8); \draw[line width=1.5] (5) to (7); \draw[line width=1.5] (5) to (8); - \draw[line width=1.5] (6) to (7); \draw[line width=1.5] (6) to (8); \draw[line width=1.5] (7) to (8); \end{scope} \end{tikzpicture} - \caption{Визуализация графа <>, $i(G) = 2, fg(G) = 3$} + \caption{Визуализация графа <>, $i(G) = 1$, $fg(G) = 1$} \end{figure} - \begin{figure}[h] \centering \begin{tikzpicture}[scale=4] @@ -965,9 +938,8 @@ $fg(G)$. \draw[line width=1.5] (7) to (8); \end{scope} \end{tikzpicture} - \caption{Визуализация графа <>, $i(G) = 2, fg(G) = 2$} + \caption{Визуализация графа <>, $i(G) = 2$, $fg(G) = 2$} \end{figure} - \begin{figure}[h] \centering \begin{tikzpicture}[scale=4] @@ -982,26 +954,37 @@ $fg(G)$. (0.174, -0.985) node[line width=1.5, draw=black, circle] (7){7} (0.766, -0.643) node[line width=1.5, draw=black, circle] (8){8}; \begin{scope}[-] + \draw[line width=1.5] (0) to (2); + \draw[line width=1.5] (0) to (4); + \draw[line width=1.5] (0) to (5); \draw[line width=1.5] (0) to (6); \draw[line width=1.5] (0) to (7); \draw[line width=1.5] (0) to (8); + \draw[line width=1.5] (1) to (3); + \draw[line width=1.5] (1) to (4); + \draw[line width=1.5] (1) to (6); \draw[line width=1.5] (1) to (7); \draw[line width=1.5] (1) to (8); + \draw[line width=1.5] (2) to (4); + \draw[line width=1.5] (2) to (5); + \draw[line width=1.5] (2) to (6); \draw[line width=1.5] (2) to (7); \draw[line width=1.5] (2) to (8); + \draw[line width=1.5] (3) to (5); + \draw[line width=1.5] (3) to (6); \draw[line width=1.5] (3) to (7); - \draw[line width=1.5] (3) to (8); - \draw[line width=1.5] (4) to (7); + \draw[line width=1.5] (4) to (5); + \draw[line width=1.5] (4) to (6); \draw[line width=1.5] (4) to (8); \draw[line width=1.5] (5) to (7); \draw[line width=1.5] (5) to (8); + \draw[line width=1.5] (6) to (7); \draw[line width=1.5] (6) to (8); \draw[line width=1.5] (7) to (8); \end{scope} \end{tikzpicture} - \caption{Визуализация графа <>, $i(G) = 6, fg(G) = 1$} + \caption{Визуализация графа <>, $i(G) = 2$, $fg(G) = 3$} \end{figure} - \begin{figure}[h] \centering \begin{tikzpicture}[scale=4] @@ -1019,9 +1002,8 @@ $fg(G)$. \begin{scope}[-] \end{scope} \end{tikzpicture} - \caption{Визуализация графа <>, $i(G) = 10, fg(G) = 0$} + \caption{Визуализация графа <>, $i(G) = 10$, $fg(G) = 0$} \end{figure} - \begin{figure}[h] \centering \begin{tikzpicture}[scale=4] @@ -1037,25 +1019,23 @@ $fg(G)$. (0.309, -0.951) node[line width=1.5, draw=black, circle] (8){8} (0.809, -0.588) node[line width=1.5, draw=black, circle] (9){9}; \begin{scope}[-] - \draw[line width=1.5] (0) to (5); - \draw[line width=1.5] (0) to (6); \draw[line width=1.5] (0) to (7); - \draw[line width=1.5] (1) to (5); + \draw[line width=1.5] (0) to (8); + \draw[line width=1.5] (0) to (9); + \draw[line width=1.5] (1) to (7); \draw[line width=1.5] (1) to (8); - \draw[line width=1.5] (2) to (6); - \draw[line width=1.5] (2) to (7); - \draw[line width=1.5] (2) to (8); - \draw[line width=1.5] (3) to (8); - \draw[line width=1.5] (5) to (7); + \draw[line width=1.5] (1) to (9); + \draw[line width=1.5] (2) to (9); + \draw[line width=1.5] (3) to (9); + \draw[line width=1.5] (4) to (9); \draw[line width=1.5] (5) to (9); \draw[line width=1.5] (6) to (9); \draw[line width=1.5] (7) to (9); \draw[line width=1.5] (8) to (9); \end{scope} \end{tikzpicture} - \caption{Визуализация графа <>, $i(G) = 4, fg(G) = 0$} + \caption{Визуализация графа <>, $i(G) = 1$, $fg(G) = 2$} \end{figure} - \begin{figure}[h] \centering \begin{tikzpicture}[scale=4] @@ -1071,25 +1051,25 @@ $fg(G)$. (0.309, -0.951) node[line width=1.5, draw=black, circle] (8){8} (0.809, -0.588) node[line width=1.5, draw=black, circle] (9){9}; \begin{scope}[-] - \draw[line width=1.5] (0) to (5); - \draw[line width=1.5] (0) to (7); + \draw[line width=1.5] (0) to (6); \draw[line width=1.5] (0) to (8); - \draw[line width=1.5] (1) to (5); + \draw[line width=1.5] (1) to (7); \draw[line width=1.5] (1) to (8); - \draw[line width=1.5] (2) to (6); - \draw[line width=1.5] (3) to (6); - \draw[line width=1.5] (4) to (7); + \draw[line width=1.5] (2) to (8); + \draw[line width=1.5] (2) to (9); + \draw[line width=1.5] (3) to (8); + \draw[line width=1.5] (3) to (9); \draw[line width=1.5] (4) to (8); - \draw[line width=1.5] (5) to (7); + \draw[line width=1.5] (4) to (9); + \draw[line width=1.5] (5) to (8); \draw[line width=1.5] (5) to (9); \draw[line width=1.5] (6) to (9); \draw[line width=1.5] (7) to (9); \draw[line width=1.5] (8) to (9); \end{scope} \end{tikzpicture} - \caption{Визуализация графа <>, $i(G) = 4, fg(G) = 1$} + \caption{Визуализация графа <>, $i(G) = 3$, $fg(G) = 2$} \end{figure} - \begin{figure}[h] \centering \begin{tikzpicture}[scale=4] @@ -1105,33 +1085,20 @@ $fg(G)$. (0.309, -0.951) node[line width=1.5, draw=black, circle] (8){8} (0.809, -0.588) node[line width=1.5, draw=black, circle] (9){9}; \begin{scope}[-] - \draw[line width=1.5] (0) to (3); - \draw[line width=1.5] (0) to (6); - \draw[line width=1.5] (0) to (7); + \draw[line width=1.5] (0) to (5); \draw[line width=1.5] (0) to (8); - \draw[line width=1.5] (1) to (4); - \draw[line width=1.5] (1) to (5); \draw[line width=1.5] (1) to (6); - \draw[line width=1.5] (1) to (7); - \draw[line width=1.5] (2) to (4); - \draw[line width=1.5] (2) to (5); + \draw[line width=1.5] (1) to (8); + \draw[line width=1.5] (1) to (9); \draw[line width=1.5] (2) to (7); - \draw[line width=1.5] (2) to (8); - \draw[line width=1.5] (3) to (6); + \draw[line width=1.5] (2) to (9); \draw[line width=1.5] (3) to (8); - \draw[line width=1.5] (4) to (5); - \draw[line width=1.5] (4) to (7); \draw[line width=1.5] (4) to (9); \draw[line width=1.5] (5) to (9); - \draw[line width=1.5] (6) to (8); - \draw[line width=1.5] (6) to (9); - \draw[line width=1.5] (7) to (9); - \draw[line width=1.5] (8) to (9); \end{scope} \end{tikzpicture} - \caption{Визуализация графа <>, $i(G) = 2, fg(G) = 1$} + \caption{Визуализация графа <>, $i(G) = 4$, $fg(G) = 1$} \end{figure} - \begin{figure}[h] \centering \begin{tikzpicture}[scale=4] @@ -1148,20 +1115,29 @@ $fg(G)$. (0.809, -0.588) node[line width=1.5, draw=black, circle] (9){9}; \begin{scope}[-] \draw[line width=1.5] (0) to (5); - \draw[line width=1.5] (0) to (8); + \draw[line width=1.5] (0) to (7); + \draw[line width=1.5] (0) to (9); \draw[line width=1.5] (1) to (6); + \draw[line width=1.5] (1) to (7); \draw[line width=1.5] (1) to (8); \draw[line width=1.5] (1) to (9); - \draw[line width=1.5] (2) to (7); + \draw[line width=1.5] (2) to (6); + \draw[line width=1.5] (2) to (8); \draw[line width=1.5] (2) to (9); - \draw[line width=1.5] (3) to (8); + \draw[line width=1.5] (3) to (7); + \draw[line width=1.5] (3) to (9); + \draw[line width=1.5] (4) to (8); \draw[line width=1.5] (4) to (9); + \draw[line width=1.5] (5) to (8); \draw[line width=1.5] (5) to (9); + \draw[line width=1.5] (6) to (7); + \draw[line width=1.5] (6) to (9); + \draw[line width=1.5] (7) to (8); + \draw[line width=1.5] (8) to (9); \end{scope} \end{tikzpicture} - \caption{Визуализация графа <>, $i(G) = 5, fg(G) = 1$} + \caption{Визуализация графа <>, $i(G) = 2$, $fg(G) = 4$} \end{figure} - \begin{figure}[h] \centering \begin{tikzpicture}[scale=4] @@ -1178,30 +1154,23 @@ $fg(G)$. (0.809, -0.588) node[line width=1.5, draw=black, circle] (9){9}; \begin{scope}[-] \draw[line width=1.5] (0) to (5); + \draw[line width=1.5] (0) to (6); \draw[line width=1.5] (0) to (7); - \draw[line width=1.5] (0) to (9); - \draw[line width=1.5] (1) to (6); - \draw[line width=1.5] (1) to (7); + \draw[line width=1.5] (1) to (5); \draw[line width=1.5] (1) to (8); - \draw[line width=1.5] (1) to (9); \draw[line width=1.5] (2) to (6); + \draw[line width=1.5] (2) to (7); \draw[line width=1.5] (2) to (8); - \draw[line width=1.5] (2) to (9); - \draw[line width=1.5] (3) to (7); - \draw[line width=1.5] (3) to (9); - \draw[line width=1.5] (4) to (8); - \draw[line width=1.5] (4) to (9); - \draw[line width=1.5] (5) to (8); + \draw[line width=1.5] (3) to (8); + \draw[line width=1.5] (5) to (7); \draw[line width=1.5] (5) to (9); - \draw[line width=1.5] (6) to (7); \draw[line width=1.5] (6) to (9); - \draw[line width=1.5] (7) to (8); + \draw[line width=1.5] (7) to (9); \draw[line width=1.5] (8) to (9); \end{scope} \end{tikzpicture} - \caption{Визуализация графа <>, $i(G) = 5, fg(G) = 4$} + \caption{Визуализация графа <>, $i(G) = 3$, $fg(G) = 0$} \end{figure} - \begin{figure}[h] \centering \begin{tikzpicture}[scale=4] @@ -1218,30 +1187,23 @@ $fg(G)$. (0.809, -0.588) node[line width=1.5, draw=black, circle] (9){9}; \begin{scope}[-] \draw[line width=1.5] (0) to (5); - \draw[line width=1.5] (0) to (6); + \draw[line width=1.5] (0) to (7); \draw[line width=1.5] (0) to (8); \draw[line width=1.5] (1) to (5); - \draw[line width=1.5] (1) to (7); \draw[line width=1.5] (1) to (8); - \draw[line width=1.5] (2) to (5); - \draw[line width=1.5] (2) to (7); - \draw[line width=1.5] (2) to (9); + \draw[line width=1.5] (2) to (6); \draw[line width=1.5] (3) to (6); - \draw[line width=1.5] (3) to (7); - \draw[line width=1.5] (3) to (9); - \draw[line width=1.5] (4) to (6); + \draw[line width=1.5] (4) to (7); \draw[line width=1.5] (4) to (8); - \draw[line width=1.5] (4) to (9); \draw[line width=1.5] (5) to (7); - \draw[line width=1.5] (5) to (8); - \draw[line width=1.5] (6) to (8); + \draw[line width=1.5] (5) to (9); \draw[line width=1.5] (6) to (9); \draw[line width=1.5] (7) to (9); + \draw[line width=1.5] (8) to (9); \end{scope} \end{tikzpicture} - \caption{Визуализация графа <>, $i(G) = 5, fg(G) = 5$} + \caption{Визуализация графа <>, $i(G) = 3$, $fg(G) = 1$} \end{figure} - \begin{figure}[h] \centering \begin{tikzpicture}[scale=4] @@ -1257,24 +1219,30 @@ $fg(G)$. (0.309, -0.951) node[line width=1.5, draw=black, circle] (8){8} (0.809, -0.588) node[line width=1.5, draw=black, circle] (9){9}; \begin{scope}[-] - \draw[line width=1.5] (0) to (7); + \draw[line width=1.5] (0) to (5); + \draw[line width=1.5] (0) to (6); \draw[line width=1.5] (0) to (8); - \draw[line width=1.5] (0) to (9); + \draw[line width=1.5] (1) to (5); \draw[line width=1.5] (1) to (7); \draw[line width=1.5] (1) to (8); - \draw[line width=1.5] (1) to (9); + \draw[line width=1.5] (2) to (5); + \draw[line width=1.5] (2) to (7); \draw[line width=1.5] (2) to (9); + \draw[line width=1.5] (3) to (6); + \draw[line width=1.5] (3) to (7); \draw[line width=1.5] (3) to (9); + \draw[line width=1.5] (4) to (6); + \draw[line width=1.5] (4) to (8); \draw[line width=1.5] (4) to (9); - \draw[line width=1.5] (5) to (9); + \draw[line width=1.5] (5) to (7); + \draw[line width=1.5] (5) to (8); + \draw[line width=1.5] (6) to (8); \draw[line width=1.5] (6) to (9); \draw[line width=1.5] (7) to (9); - \draw[line width=1.5] (8) to (9); \end{scope} \end{tikzpicture} - \caption{Визуализация графа <>, $i(G) = 7, fg(G) = 2$} + \caption{Визуализация графа <>, $i(G) = 2$, $fg(G) = 5$} \end{figure} - \begin{figure}[h] \centering \begin{tikzpicture}[scale=4] @@ -1290,26 +1258,32 @@ $fg(G)$. (0.309, -0.951) node[line width=1.5, draw=black, circle] (8){8} (0.809, -0.588) node[line width=1.5, draw=black, circle] (9){9}; \begin{scope}[-] + \draw[line width=1.5] (0) to (3); \draw[line width=1.5] (0) to (6); + \draw[line width=1.5] (0) to (7); \draw[line width=1.5] (0) to (8); + \draw[line width=1.5] (1) to (4); + \draw[line width=1.5] (1) to (5); + \draw[line width=1.5] (1) to (6); \draw[line width=1.5] (1) to (7); - \draw[line width=1.5] (1) to (8); + \draw[line width=1.5] (2) to (4); + \draw[line width=1.5] (2) to (5); + \draw[line width=1.5] (2) to (7); \draw[line width=1.5] (2) to (8); - \draw[line width=1.5] (2) to (9); + \draw[line width=1.5] (3) to (6); \draw[line width=1.5] (3) to (8); - \draw[line width=1.5] (3) to (9); - \draw[line width=1.5] (4) to (8); + \draw[line width=1.5] (4) to (5); + \draw[line width=1.5] (4) to (7); \draw[line width=1.5] (4) to (9); - \draw[line width=1.5] (5) to (8); \draw[line width=1.5] (5) to (9); + \draw[line width=1.5] (6) to (8); \draw[line width=1.5] (6) to (9); \draw[line width=1.5] (7) to (9); \draw[line width=1.5] (8) to (9); \end{scope} \end{tikzpicture} - \caption{Визуализация графа <>, $i(G) = 6, fg(G) = 2$} + \caption{Визуализация графа <>, $i(G) = 2$, $fg(G) = 1$} \end{figure} - \begin{figure}[h] \centering \begin{tikzpicture}[scale=4] @@ -1353,7 +1327,7 @@ $fg(G)$. \draw[line width=1.5] (8) to (9); \end{scope} \end{tikzpicture} - \caption{Визуализация графа <>, $i(G) = 2, fg(G) = 3$} + \caption{Визуализация графа <>, $i(G) = 2$, $fg(G) = 3$} \end{figure} \conclusion -- cgit v1.2.3