diff --git a/papers/plane_depth_sequencing/paper.aux b/papers/plane_depth_sequencing/paper.aux index d5dfff2..dcd0a21 100644 --- a/papers/plane_depth_sequencing/paper.aux +++ b/papers/plane_depth_sequencing/paper.aux @@ -1,8 +1,11 @@ \relax +\citation{baker1994} \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Definitions}}{1}{}\protected@file@percent } +\bibcite{baker1994}{1} \newlabel{tocindent-1}{0pt} -\newlabel{tocindent0}{0pt} +\newlabel{tocindent0}{12.7778pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{2}{}\protected@file@percent } \gdef \@abspage@last{2} diff --git a/papers/plane_depth_sequencing/paper.fdb_latexmk b/papers/plane_depth_sequencing/paper.fdb_latexmk index b6f68d1..47b7adf 100644 --- a/papers/plane_depth_sequencing/paper.fdb_latexmk +++ b/papers/plane_depth_sequencing/paper.fdb_latexmk @@ -1,6 +1,6 @@ # Fdb version 4 -["pdflatex"] 1777104894.61083 "/home/didericis/Code/math-research/papers/plane_depth_sequencing/paper.tex" "paper.pdf" "paper" 1777104894.88704 0 - "/home/didericis/Code/math-research/papers/plane_depth_sequencing/paper.tex" 1777104894.41564 5298 4a3acc81f8f9c682ff322e02fe18ccf7 "" +["pdflatex"] 1777106346.98886 "/home/didericis/Code/math-research/papers/plane_depth_sequencing/paper.tex" "paper.pdf" "paper" 1777106347.2627 0 + "/home/didericis/Code/math-research/papers/plane_depth_sequencing/paper.tex" 1777106346.52452 7702 1354320d93134a233d8c3c035e09f7c0 "" "/nix/store/4g7bv3lsd1r7lrfxi0x145xac0jag4hl-texlive-combined-full-2025.20250703/share/texmf-var/fonts/map/pdftex/updmap/pdftex.map" 1 5523663 ec1f96d89b308e150332b305019a3402 "" "/nix/store/4g7bv3lsd1r7lrfxi0x145xac0jag4hl-texlive-combined-full-2025.20250703/share/texmf-var/web2c/pdftex/pdflatex.fmt" 1 3600504 177ced77725200f4fa24b79427ded12f "" "/nix/store/4g7bv3lsd1r7lrfxi0x145xac0jag4hl-texlive-combined-full-2025.20250703/share/texmf-var/web2c/texmf.cnf" 1 44455 00ca67f5a06c9c23b32559f3f48cb4e9 "" @@ -45,8 +45,8 @@ "/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsmath/amsopn.sty" 1 4474 c510a88aa5f51b8c773b50a7ee92befd "" "/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsmath/amstext.sty" 1 2444 9983e1d0683f102e3b190c64a49313aa "" "/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/l3backend/l3backend-pdftex.def" 1 30351 a2b09edc6c93a742566b222c33d0278e "" - "paper.aux" 1777104894.84465 278 6ee4d5d6b3925666e01309d058e35079 "pdflatex" - "paper.tex" 1777104894.41564 5298 4a3acc81f8f9c682ff322e02fe18ccf7 "" + "paper.aux" 1777106347.22354 429 e867892bc6d7fe276fbd19bcc4f1bc53 "pdflatex" + "paper.tex" 1777106346.52452 7702 1354320d93134a233d8c3c035e09f7c0 "" (generated) "paper.aux" "paper.log" diff --git a/papers/plane_depth_sequencing/paper.log b/papers/plane_depth_sequencing/paper.log index bf47b9d..4074572 100644 --- a/papers/plane_depth_sequencing/paper.log +++ b/papers/plane_depth_sequencing/paper.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.27 (TeX Live 2025/nixos.org) (preloaded format=pdflatex 1980.1.1) 25 APR 2026 04:15 +This is pdfTeX, Version 3.141592653-2.6-1.40.27 (TeX Live 2025/nixos.org) (preloaded format=pdflatex 1980.1.1) 25 APR 2026 04:39 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -176,10 +176,10 @@ L3 programming layer <2025-06-09> *********** ) Here is how much of TeX's memory you used: - 1763 strings out of 467888 - 25768 string characters out of 5405403 - 435018 words of memory out of 5000000 - 30194 multiletter control sequences out of 15000+600000 + 1768 strings out of 467888 + 25909 string characters out of 5405403 + 437018 words of memory out of 5000000 + 30199 multiletter control sequences out of 15000+600000 633232 words of font info for 65 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 71i,6n,79p,751b,236s stack positions out of 10000i,1000n,20000p,200000b,200000s @@ -204,7 +204,7 @@ i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts /type1/public/amsfonts/cm/cmti8.pfb> -Output written on paper.pdf (2 pages, 144144 bytes). +Output written on paper.pdf (2 pages, 158297 bytes). PDF statistics: 71 PDF objects out of 1000 (max. 8388607) 42 compressed objects within 1 object stream diff --git a/papers/plane_depth_sequencing/paper.pdf b/papers/plane_depth_sequencing/paper.pdf index c4b22bd..a2d1293 100644 Binary files a/papers/plane_depth_sequencing/paper.pdf and b/papers/plane_depth_sequencing/paper.pdf differ diff --git a/papers/plane_depth_sequencing/paper.tex b/papers/plane_depth_sequencing/paper.tex index 6559ed3..6293420 100644 --- a/papers/plane_depth_sequencing/paper.tex +++ b/papers/plane_depth_sequencing/paper.tex @@ -94,6 +94,30 @@ An edge $\{u, v\} \in E(G)$ is a \emph{level edge} if $\mathrm{depth}(u) = \math A triangle $\{u, v, w\}$ in $G$ is an \emph{up triangle} if the multiset of depths of its vertices is $\{d, d+1, d+1\}$ for some $d \geq 0$, a \emph{down triangle} if the multiset of depths is $\{d, d, d+1\}$ for some $d \geq 0$, and a \emph{neutral triangle} if the multiset of depths is $\{d, d, d\}$ for some $d \geq 0$. \end{definition} +\begin{remark} +We now relate our terminology to existing terminology, namely $k$-outerplanar graphs \cite{baker1994}. The following definition and lemma show that the subgraph induced by any single depth level is outerplanar, i.e., $1$-outerplanar in the sense of Baker. +\end{remark} + +\begin{definition} +A plane graph is \emph{outerplanar} if every vertex lies on the outer face. More generally, a plane graph is \emph{$k$-outerplanar} for $k \geq 1$ if removing all vertices on the outer face yields a $(k-1)$-outerplanar graph, where every graph on the empty vertex set is $0$-outerplanar. +\end{definition} + +\begin{lemma} +Let $G$ be a graph with a plane embedding and outer cycle $C$. For each $d \geq 0$, the subgraph of $G$ induced by $V_d = \{v \in V(G) : \mathrm{depth}(v) = d\}$ is outerplanar. +\end{lemma} + +\begin{proof} +Let $H = G[V_d]$ with the plane embedding inherited from $G$. It suffices to show every vertex of $H$ lies on the outer face of $H$. + +For $d = 0$, we have $V_0 = V(C)$ and $H$ is a subgraph of the cycle $C$, hence a disjoint union of paths, which is outerplanar. + +For $d \geq 1$, let $U$ be the open subset of the plane obtained by removing all vertices and edges of $H$. We show every $v \in V_d$ lies on the boundary of the component $U_{\mathrm{out}}$ of $U$ containing the outer face of $G$. + +Since every vertex in $V_{\leq d-1}$ has a shortest path to $C$ passing entirely through $V_{\leq d-1}$, the subgraph $G[V_{\leq d-1}]$ is connected and contains $C$. Its vertices and edges lie in $U$ (as they are not in $H$), and $C$ borders the outer face of $G$, so $G[V_{\leq d-1}]$ and the outer face of $G$ are connected within $U$, hence both lie in $U_{\mathrm{out}}$. + +Now let $v \in V_d$. Since $\mathrm{depth}(v) = d \geq 1$, there exists $u \in V_{d-1}$ adjacent to $v$ in $G$. The edge $\{v, u\}$ is not an edge of $H$, so it lies in $U$. Since $u \in V_{d-1} \subset U_{\mathrm{out}}$ and $\{v,u\}$ is a connected subset of $U$ containing $u$, the entire edge lies in $U_{\mathrm{out}}$. The vertex $v$ is an endpoint of this edge but is not in $U$, so $v$ lies on the boundary of $U_{\mathrm{out}}$, i.e., on the outer face of $H$. +\end{proof} + \begin{definition} Let $G$ be a maximal planar graph with a plane embedding and outer cycle $C$. The \emph{deep embedding} of $G$ is the graph $G'$ obtained from $G$ by the following operation: for every 3-cycle $\{u, v, w\} \subseteq V(G)$ such that \[ @@ -120,6 +144,15 @@ We now consider each case under the deep embedding. Since every face of $G'$ falls into one of these cases, the result follows. \end{proof} +\begin{thebibliography}{9} + +\bibitem{baker1994} +B.~S.~Baker, +\emph{Approximation algorithms for {NP}-complete problems on planar graphs}, +Journal of the ACM, vol.~41, no.~1, pp.~153--180, 1994. + +\end{thebibliography} + \end{document} %-----------------------------------------------------------------------