اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدThe edit distance function of some graphs
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The edit distance function of a hereditary property $mathscr{H}$ is the asymptotically largest edit distance between a graph of density $pin[0,1]$ and $mathscr{H}$. Denote by $P_n$ and $C_n$ the path graph of order $n$ and the cycle graph of order $n$, respectively. Let $C_{2n}^*$ be the cycle graph $C_{2n}$ with a diagonal, and $widetilde{C_n}$ be the graph with vertex set ${v_0, v_1, ldots, v_{n-1}}$ and $E(widetilde{C_n})=E(C_n)cup {v_0v_2}$. Marchant and Thomason determined the edit distance function of $C_6^{*}$. Peck studied the edit distance function of $C_n$, while Berikkyzy et al. studied the edit distance of powers of cycles. In this paper, by using the methods of Peck and Martin, we determine the edit distance function of $C_8^{*}$, $widetilde{C_n}$ and $P_n$, respectively.
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Given a hereditary property $mathcal H$ of graphs and some $pin[0,1]$, the edit distance function $operatorname{ed}_{mathcal H}(p)$ is (asymptotically) the maximum proportion of edits (edge-additions plus edge-deletions) necessary to transform any gr
aph of density $p$ into a member of $mathcal H$. For any fixed $pin[0,1]$, $operatorname{ed}_{mathcal H}(p)$ can be computed from an object known as a colored regularity graph (CRG). This paper is concerned with those points $pin[0,1]$ for which infinitely many CRGs are required to compute $operatorname{ed}_{mathcal H}$ on any open interval containing $p$; such a $p$ is called an accumulation point. We show that, as expected, $p=0$ and $p=1$ are indeed accumulation points for some hereditary properties; we additionally determine the slope of $operatorname{ed}_{mathcal H}$ at these two extreme points. Unexpectedly, we construct a hereditary property with an accumulation point at $p=1/4$. Finally, we derive a significant structural property about those CRGs which occur at accumulation points.
Given a hereditary property of graphs $mathcal{H}$ and a $pin [0,1]$, the edit distance function ${rm ed}_{mathcal{H}}(p)$ is asymptotically the maximum proportion of edge-additions plus edge-deletions applied to a graph of edge density $p$ sufficien
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