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[lyx.git] / lib / examples / beamerlyxexample1.lyx

diff --git a/lib/examples/beamerlyxexample1.lyx b/lib/examples/beamerlyxexample1.lyx
index 557a4fb851c7cfb8df5eda09d7c438bb7f52c212..922cee1048d4b2c31b9eb95db34ecd32d451f7c6 100644 (file)
--- a/lib/examples/beamerlyxexample1.lyx
+++ b/lib/examples/beamerlyxexample1.lyx
@@ -1,7 +1,8 @@
-#LyX 2.2 created this file. For more info see http://www.lyx.org/
-\lyxformat 501
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
 \begin_document
 \begin_header
+\save_transient_properties true
 \origin /systemlyxdir/examples/
 \textclass beamer
 \begin_preamble
@@ -156,9 +157,9 @@
 \language_package default
 \inputencoding auto
 \fontencoding global
-\font_roman "times" "default"
-\font_sans "default" "default"
-\font_typewriter "default" "default"
+\font_roman "lmodern" "default"
+\font_sans "lmss" "default"
+\font_typewriter "lmtt" "default"
 \font_math "auto" "auto"
 \font_default_family default
 \use_non_tex_fonts false
@@ -166,6 +167,8 @@
 \font_osf false
 \font_sf_scale 100 100
 \font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures false
 \graphics default
 \default_output_format default
 \output_sync 0
@@ -175,17 +178,17 @@
 \spacing single
 \use_hyperref false
 \papersize default
-\use_geometry false
+\use_geometry true
 \use_package amsmath 2
 \use_package amssymb 2
-\use_package cancel 0
-\use_package esint 0
+\use_package cancel 1
+\use_package esint 1
 \use_package mathdots 1
-\use_package mathtools 0
+\use_package mathtools 1
 \use_package mhchem 1
-\use_package stackrel 0
-\use_package stmaryrd 0
-\use_package undertilde 0
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
 \cite_engine basic
 \cite_engine_type default
 \biblio_style plain
@@ -195,6 +198,7 @@
 \suppress_date false
 \justification true
 \use_refstyle 0
+\use_minted 0
 \index Index
 \shortcut idx
 \color #008000
@@ -203,7 +207,10 @@
 \tocdepth 2
 \paragraph_separation indent
 \paragraph_indentation default
-\quotes_language english
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style english
+\dynamic_quotes 0
 \papercolumns 1
 \papersides 1
 \paperpagestyle default
@@ -261,9 +268,6 @@ Outline
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -364,17 +368,11 @@ Tournaments Consist of Jousts Between Knights
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
 \begin_layout Columns
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -563,9 +561,6 @@ What is a Tournament?
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -574,6 +569,7 @@ What is a Tournament?
 status open
 
 \begin_layout Plain Layout
+
 1-
 \end_layout
 
@@ -587,6 +583,7 @@ A group of knights.
 status open
 
 \begin_layout Plain Layout
+
 2-
 \end_layout
 
@@ -600,6 +597,7 @@ Every pair has a joust.
 status open
 
 \begin_layout Plain Layout
+
 3-
 \end_layout
 
@@ -612,12 +610,11 @@ In every joust one knight wins.
 \end_deeper
 \end_deeper
 \begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
 \end_inset
 
-\end_layout
 
+\end_layout
 
 \begin_layout Frame
 \begin_inset Argument 4
@@ -630,17 +627,11 @@ Tournaments are Complete Directed Graphs
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
 \begin_layout Columns
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -853,6 +844,7 @@ end{pgfpicture}
 status collapsed
 
 \begin_layout Plain Layout
+
 2-
 \end_layout
 
@@ -867,10 +859,6 @@ tournament
  
 \color inherit
 is a
-\color inherit
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -890,18 +878,18 @@ any two different vertices.
 \end_deeper
 \end_deeper
 \begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
 \end_inset
 
-\end_layout
 
+\end_layout
 
 \begin_layout Frame
 \begin_inset Argument 2
 status collapsed
 
 \begin_layout Plain Layout
+
 +
 \end_layout
 
@@ -918,9 +906,6 @@ Tournaments Arise Naturally in Different Situations
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -935,9 +920,6 @@ Applications in Ordering Theory
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -952,12 +934,11 @@ The comparison relation may be cyclic, however.
 
 \end_deeper
 \begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
 \end_inset
 
-\end_layout
 
+\end_layout
 
 \begin_layout ExampleBlock
 \begin_inset Argument 2
@@ -970,9 +951,6 @@ Applications in Sociology
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -987,12 +965,11 @@ Reviewers decide for any two candidates whom they prefer.
 
 \end_deeper
 \begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
 \end_inset
 
-\end_layout
 
+\end_layout
 
 \begin_layout ExampleBlock
 \begin_inset Argument 2
@@ -1005,9 +982,6 @@ Applications in Structural Complexity Theory
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -1048,9 +1022,6 @@ status open
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -1068,6 +1039,7 @@ status open
 status open
 
 \begin_layout Plain Layout
+
 1
 \end_layout
 
@@ -1087,6 +1059,7 @@ status open
 status open
 
 \begin_layout Plain Layout
+
 2-3
 \end_layout
 
@@ -1110,6 +1083,7 @@ status open
 status open
 
 \begin_layout Plain Layout
+
 4-5
 \end_layout
 
@@ -1129,6 +1103,7 @@ status open
 status open
 
 \begin_layout Plain Layout
+
 6-7
 \end_layout
 
@@ -1148,6 +1123,7 @@ status open
 status open
 
 \begin_layout Plain Layout
+
 8-9
 \end_layout
 
@@ -1171,6 +1147,7 @@ status open
 status open
 
 \begin_layout Plain Layout
+
 10-
 \end_layout
 
@@ -1187,9 +1164,6 @@ the Approximation Problem
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -1238,6 +1212,7 @@ target
 status open
 
 \begin_layout Plain Layout
+
 only@-9| visible@8-
 \end_layout
 
@@ -1278,6 +1253,7 @@ phantom{p}
 status open
 
 \begin_layout Plain Layout
+
 only@10-
 \end_layout
 
@@ -1320,15 +1296,13 @@ nointerlineskip
 status open
 
 \begin_layout Plain Layout
+
 1,3,5,7,9,11-12
 \end_layout
 
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -1343,9 +1317,6 @@ t,onlytextwidth
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -1358,6 +1329,7 @@ status open
 status open
 
 \begin_layout Plain Layout
+
 1-2
 \end_layout
 
@@ -1419,9 +1391,6 @@ Example Input
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -1764,6 +1733,7 @@ status open
 status open
 
 \begin_layout Plain Layout
+
 3-
 \end_layout
 
@@ -1795,6 +1765,7 @@ column{5cm}
 status collapsed
 
 \begin_layout Plain Layout
+
 only@3-
 \end_layout
 
@@ -1811,9 +1782,6 @@ Example Output
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -2034,15 +2002,13 @@ end{pgfpicture}
 status open
 
 \begin_layout Plain Layout
+
 2,4,6,8,10
 \end_layout
 
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -2057,9 +2023,6 @@ Variants of Path Finding Problems
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -2068,6 +2031,7 @@ Variants of Path Finding Problems
 status open
 
 \begin_layout Plain Layout
+
 2-
 \end_layout
 
@@ -2107,6 +2071,7 @@ Approximation Problem:
 status open
 
 \begin_layout Plain Layout
+
 4-
 \end_layout
 
@@ -2136,6 +2101,7 @@ Problem: Construct a path from
 status open
 
 \begin_layout Plain Layout
+
 6-
 \end_layout
 
@@ -2165,6 +2131,7 @@ Problem: Construct a shortest path from
 status open
 
 \begin_layout Plain Layout
+
 8-
 \end_layout
 
@@ -2202,6 +2169,7 @@ Problem: Is the distance of
 status open
 
 \begin_layout Plain Layout
+
 10-
 \end_layout
 
@@ -2275,9 +2243,6 @@ Logspace Turing Machines
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -2446,12 +2411,11 @@ end{pgfpicture}
 
 \end_deeper
 \begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
 \end_inset
 
-\end_layout
 
+\end_layout
 
 \begin_layout Frame
 \begin_inset Argument 4
@@ -2464,9 +2428,6 @@ Logspace Turing Machines Are Quite Powerful
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -2481,9 +2442,6 @@ Deterministic logspace machines can compute
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -2515,9 +2473,6 @@ Non-deterministic logspace machines can compute
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -2536,18 +2491,18 @@ satisfiability with two literals per clause.
 \end_deeper
 \end_deeper
 \begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
 \end_inset
 
-\end_layout
 
+\end_layout
 
 \begin_layout Frame
 \begin_inset Argument 1
 status collapsed
 
 \begin_layout Plain Layout
+
 1
 \end_layout
 
@@ -2574,9 +2529,6 @@ The Complexity Class Hierarchy
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -2965,12 +2917,11 @@ end{pgfpicture}
 
 \end_deeper
 \begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
 \end_inset
 
-\end_layout
 
+\end_layout
 
 \begin_layout Frame
 \begin_inset Argument 4
@@ -2999,9 +2950,6 @@ Limit the Circuit Depth
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -3041,9 +2989,6 @@ t
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -3066,9 +3011,6 @@ Circuit Class
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -3086,9 +3028,6 @@ unbounded fan-in
 \end_deeper
 \begin_layout Examples
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -3130,9 +3069,6 @@ Circuit Class
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -3150,9 +3086,6 @@ bounded fan-in
 \end_deeper
 \begin_layout Examples
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -3201,9 +3134,6 @@ Circuit Class
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -3221,9 +3151,6 @@ bounded fan-in
 \end_deeper
 \begin_layout Examples
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -3242,6 +3169,7 @@ bounded fan-in
 status collapsed
 
 \begin_layout Plain Layout
+
 2
 \end_layout
 
@@ -3269,9 +3197,6 @@ Are Equally Difficult
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -3297,9 +3222,6 @@ Are Equally Difficult
 
 \begin_layout Corollary
 For directed graphs, we can solve
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -3342,6 +3264,7 @@ the approximation problem in logspace iff
 status collapsed
 
 \begin_layout Plain Layout
+
 3
 \end_layout
 
@@ -3365,9 +3288,6 @@ But Not Trivial
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -3387,12 +3307,11 @@ But Not Trivial
 \end_layout
 
 \begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
 \end_inset
 
-\end_layout
 
+\end_layout
 
 \begin_layout Fact
 \begin_inset Formula $\Lang{reach}_{\operatorname{path}}$
@@ -3415,6 +3334,7 @@ But Not Trivial
 status collapsed
 
 \begin_layout Plain Layout
+
 4
 \end_layout
 
@@ -3442,9 +3362,6 @@ Definition of the Tournament Reachability Problem
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -3466,10 +3383,6 @@ contain all triples
 \end_inset
 
  such that
-\color inherit
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -3503,12 +3416,11 @@ there exists a path from
 \end_deeper
 \end_deeper
 \begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
 \end_inset
 
-\end_layout
 
+\end_layout
 
 \begin_layout Frame
 \begin_inset Argument 4
@@ -3521,9 +3433,6 @@ The Tournament Reachability Problem is Very Easy
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -3549,9 +3458,6 @@ Implications
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -3589,6 +3495,7 @@ easier
 status open
 
 \begin_layout Plain Layout
+
 5
 \end_layout
 
@@ -3616,9 +3523,6 @@ the Distance Problem
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -3640,10 +3544,6 @@ contain all tuples
 \end_inset
 
  such that 
-\color inherit
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -3681,12 +3581,11 @@ the distance of
 \end_deeper
 \end_deeper
 \begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
 \end_inset
 
-\end_layout
 
+\end_layout
 
 \begin_layout Frame
 \begin_inset Argument 4
@@ -3699,9 +3598,6 @@ The Tournament Distance Problem is Hard
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -3767,12 +3663,11 @@ in logarithmic space, iff
 
 \end_deeper
 \begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
 \end_inset
 
-\end_layout
 
+\end_layout
 
 \begin_layout Frame
 \begin_inset Argument 4
@@ -3789,9 +3684,6 @@ Proof That
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -3822,9 +3714,6 @@ t,onlytextwidth
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -3869,9 +3758,6 @@ Reduce
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -3880,6 +3766,7 @@ Reduce
 status open
 
 \begin_layout Plain Layout
+
 alert@1
 \end_layout
 
@@ -3901,6 +3788,7 @@ Is input
 status open
 
 \begin_layout Plain Layout
+
 2-| alert@2-8
 \end_layout
 
@@ -3922,6 +3810,7 @@ Map
 status open
 
 \begin_layout Plain Layout
+
 9-| alert@9
 \end_layout
 
@@ -3940,12 +3829,11 @@ Query:
 
 \end_deeper
 \begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
 \end_inset
 
-\end_layout
 
+\end_layout
 
 \begin_layout Block
 \begin_inset Argument 2
@@ -3962,15 +3850,13 @@ Correctness
 status open
 
 \begin_layout Plain Layout
+
 10-
 \end_layout
 
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -3979,6 +3865,7 @@ status open
 status open
 
 \begin_layout Plain Layout
+
 10-| alert@10-11
 \end_layout
 
@@ -4012,6 +3899,7 @@ a length-3 path in
 status open
 
 \begin_layout Plain Layout
+
 12-| alert@12-13
 \end_layout
 
@@ -5103,6 +4991,7 @@ end{pgfpicture}
 status open
 
 \begin_layout Plain Layout
+
 6
 \end_layout
 
@@ -5126,9 +5015,6 @@ Approximators Compute Paths that Are Nearly As Short As a Shortest Path
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -5146,10 +5032,6 @@ approximation scheme for
  
 \color inherit
 gets as input
-\color inherit
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -5196,12 +5078,11 @@ a path from
 \end_deeper
 \end_deeper
 \begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
 \end_inset
 
-\end_layout
 
+\end_layout
 
 \begin_layout Frame
 \begin_inset Argument 4
@@ -5218,9 +5099,6 @@ the Tournament Shortest Path Problem
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -5280,6 +5158,7 @@ beamergotobutton{More Details}}
 status collapsed
 
 \begin_layout Plain Layout
+
 7
 \end_layout
 
@@ -5302,7 +5181,7 @@ fragile
 \end_inset
 
 .
- \SpecialCharNoPassThru LyX
+ \SpecialChar LyX
  has the FragileFrame layout for this.
 \end_layout
 
@@ -5326,9 +5205,6 @@ Just a frame with a program code listing
 
 \begin_layout FragileFrame
 This is some program code:
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -5408,9 +5284,6 @@ Summary
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -5425,9 +5298,6 @@ Summary
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -5498,12 +5368,11 @@ in tournaments is
 
 \end_deeper
 \begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
 \end_inset
 
-\end_layout
 
+\end_layout
 
 \begin_layout Block
 \begin_inset Argument 2
@@ -5516,9 +5385,6 @@ Outlook
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -5593,9 +5459,6 @@ For Further Reading
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -5619,6 +5482,7 @@ beamertemplatebookbibitems
 \begin_inset CommandInset bibitem
 LatexCommand bibitem
 key "Moon1968"
+literal "true"
 
 \end_inset
 
@@ -5679,6 +5543,7 @@ beamertemplatearticlebibitems
 \begin_inset CommandInset bibitem
 LatexCommand bibitem
 key "NickelsenT2002"
+literal "true"
 
 \end_inset
 
@@ -5725,6 +5590,7 @@ Proc.
 \begin_inset CommandInset bibitem
 LatexCommand bibitem
 key "Tantau2004b"
+literal "true"
 
 \end_inset
 
@@ -5828,9 +5694,6 @@ Definition of Independence Number of a Graph
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -5865,12 +5728,11 @@ Tournaments have independence number 1.
 
 \end_deeper
 \begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
 \end_inset
 
-\end_layout
 
+\end_layout
 
 \begin_layout Frame
 \begin_inset Argument 4
@@ -5887,9 +5749,6 @@ Graphs With Bounded Independence Number
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -5930,12 +5789,11 @@ at most
 \end_layout
 
 \begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
 \end_inset
 
-\end_layout
 
+\end_layout
 
 \begin_layout Theorem
 For each
@@ -5967,12 +5825,11 @@ for approximating the shortest path in graphs with independence number at
 \end_layout
 
 \begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
 \end_inset
 
-\end_layout
 
+\end_layout
 
 \begin_layout Theorem
 For each
@@ -6022,6 +5879,7 @@ Finding Paths in Undirected Graphs
 status collapsed
 
 \begin_layout Plain Layout
+
 1-2
 \end_layout
 
@@ -6052,9 +5910,6 @@ Is Party Unknown.
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -6071,9 +5926,6 @@ Is Party Unknown.
 
 \begin_layout Corollary
 For undirected graphs, we can solve
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -6095,6 +5947,7 @@ status open
 status open
 
 \begin_layout Plain Layout
+
 1
 \end_layout
 
@@ -6141,6 +5994,7 @@ status open
 status open
 
 \begin_layout Plain Layout
+
 1
 \end_layout
 
@@ -6209,9 +6063,6 @@ The Approximation Scheme is Optimal
 \end_inset
 
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper
@@ -6234,9 +6085,6 @@ Suppose there exists an approximation scheme for
 
 \begin_layout Proof
 
-\begin_inset Separator parbreak
-\end_inset
-
 \end_layout
 
 \begin_deeper