1 files changed, 1779 insertions, 0 deletions

diff --git a/šola/ds2/kolokvij1.lyx b/šola/ds2/kolokvij1.lyx
new file mode 100644
index 0000000..2227d5e
--- /dev/null
+++ b/šola/ds2/kolokvij1.lyx
@@ -0,0 +1,1779 @@
+#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 unavailable
+\textclass article
+\begin_preamble
+\usepackage{siunitx}
+\usepackage{pgfplots}
+\usepackage{listings}
+\usepackage{multicol}
+\sisetup{output-decimal-marker = {,}, quotient-mode=fraction, output-exponent-marker=\ensuremath{\mathrm{3}}}
+\DeclareMathOperator{\g}{g}
+\DeclareMathOperator{\sled}{sled}
+\DeclareMathOperator{\Aut}{Aut}
+\DeclareMathOperator{\Cir}{Cir}
+\DeclareMathOperator{\ecc}{ecc}
+\DeclareMathOperator{\rad}{rad}
+\DeclareMathOperator{\diam}{diam}
+\newcommand\euler{e}
+\end_preamble
+\use_default_options true
+\begin_modules
+enumitem
+\end_modules
+\maintain_unincluded_children false
+\language slovene
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics xetex
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry true
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification false
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 1cm
+\topmargin 1cm
+\rightmargin 1cm
+\bottommargin 2cm
+\headheight 1cm
+\headsep 1cm
+\footskip 1cm
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style german
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+setlength{
+\backslash
+columnseprule}{0.2pt}
+\backslash
+begin{multicols}{2}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Največja stopnja 
+\begin_inset Formula $\Delta G$
+\end_inset
+
+, najmanjša 
+\begin_inset Formula $\delta G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Rokovanje: 
+\begin_inset Formula $\sum_{v\in VG}\deg_{G}v=2\left|EG\right|$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Vsak graf ima sodo mnogo vozlišč lihe stopnje.
+\end_layout
+
+\begin_layout Standard
+Presek/unija grafov je presek/unija njunih 
+\begin_inset Formula $V$
+\end_inset
+
+ in 
+\begin_inset Formula $E$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G\cup H$
+\end_inset
+
+ je disjunktna unija grafov 
+\begin_inset Formula $\sim$
+\end_inset
+
+ 
+\begin_inset Formula $VG\cap VH=\emptyset$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Komplement grafa: 
+\begin_inset Formula $\overline{G}$
+\end_inset
+
+ (obratna povezanost)
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $0\leq\left|EG\right|\leq{\left|VG\right| \choose 2}\quad\quad\quad$
+\end_inset
+
+Za padajoče 
+\begin_inset Formula $d_{i}$
+\end_inset
+
+ velja:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left(d_{1},\dots d_{n}\right)$
+\end_inset
+
+ graf 
+\begin_inset Formula $\Leftrightarrow\left(d_{2}-1,\dots,d_{d_{1}+1}-1,\dots,d_{n}\right)$
+\end_inset
+
+ graf
+\end_layout
+
+\begin_layout Standard
+Če je 
+\begin_inset Formula $AG$
+\end_inset
+
+ matrika sosednosti, 
+\begin_inset Formula $\left(\left(AG\right)^{n}\right)_{i,j}$
+\end_inset
+
+ pove št.
+ 
+\begin_inset Formula $i,j-$
+\end_inset
+
+poti.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\text{Število trikotnikov: }\frac{\sled\left(\left(AG\right)^{3}\right)}{2\cdot3}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Sprehod
+\end_layout
+
+\begin_layout Standard
+je zaporedje vozlišč, ki so verižno povezana.
+\end_layout
+
+\begin_layout Standard
+Dolžina sprehoda je število prehojenih povezav.
+\end_layout
+
+\begin_layout Standard
+Sklenjen sprehod dolžine 
+\begin_inset Formula $k$
+\end_inset
+
+: 
+\begin_inset Formula $v_{0}=v_{k}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Enostaven sprehod ima disjunktna vozlišča razen 
+\begin_inset Formula $\left(v_{0},v_{k}\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Pot v grafu: podgraf 
+\begin_inset Formula $P_{k}$
+\end_inset
+
+ 
+\begin_inset Formula $\sim$
+\end_inset
+
+ enostaven nesklenjen sprehod.
+\end_layout
+
+\begin_layout Paragraph
+Cikel
+\end_layout
+
+\begin_layout Standard
+podgraf, ki je enostaven sklenjen sprehod dolžine 
+\begin_inset Formula $>3$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Če v 
+\begin_inset Formula $G$
+\end_inset
+
+ 
+\begin_inset Formula $\exists$
+\end_inset
+
+ dve različni 
+\begin_inset Formula $u,v-$
+\end_inset
+
+poti 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ 
+\begin_inset Formula $G$
+\end_inset
+
+ premore cikel
+\end_layout
+
+\begin_layout Standard
+Sklenjen sprehod lihe dolžine 
+\begin_inset Formula $\in G$
+\end_inset
+
+ 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ lih cikel 
+\begin_inset Formula $\in G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Povezanost
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $u,v$
+\end_inset
+
+ sta v isti komponenti 
+\begin_inset Formula $\text{\ensuremath{\sim}}$
+\end_inset
+
+ 
+\begin_inset Formula $\text{\ensuremath{\exists}}$
+\end_inset
+
+ 
+\begin_inset Formula $u,v-$
+\end_inset
+
+pot
+\end_layout
+
+\begin_layout Standard
+Število komponent grafa: 
+\begin_inset Formula $\Omega G$
+\end_inset
+
+.
+ 
+\begin_inset Formula $G$
+\end_inset
+
+ povezan 
+\begin_inset Formula $\sim\Omega G=1$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Komponenta je maksimalen povezan podgraf.
+\end_layout
+
+\begin_layout Standard
+Premer: 
+\begin_inset Formula $\diam G=\max\left\{ d_{G}\left(v,u\right);\forall v,u\in VG\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Ekscentričnost: 
+\begin_inset Formula $\ecc_{G}u=max\left\{ d_{G}\left(u,x\right);\forall x\in VG\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Polmer: 
+\begin_inset Formula $\rad G=\min\left\{ \ecc u;\forall u\in VG\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\diam C_{n}=\rad C_{n}=\lfloor\frac{n}{2}\rfloor$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+(Liha) ožina (
+\begin_inset Formula $\g G$
+\end_inset
+
+) je dolžina najkrajšega (lihega) cikla.
+\end_layout
+
+\begin_layout Standard
+Vsaj en od 
+\begin_inset Formula $G$
+\end_inset
+
+ in 
+\begin_inset Formula $\overline{G}$
+\end_inset
+
+ je povezan.
+\end_layout
+
+\begin_layout Standard
+Povezava 
+\begin_inset Formula $e$
+\end_inset
+
+ je most 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ 
+\begin_inset Formula $\Omega\left(G-e\right)>\Omega G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $u$
+\end_inset
+
+ je prerezno vozlišče 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ 
+\begin_inset Formula $\Omega\left(G-u\right)>\Omega G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Za nepovezan 
+\begin_inset Formula $G$
+\end_inset
+
+ velja 
+\begin_inset Formula $\diam\overline{G}\leq2$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Dvodelni
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\sim V=A\cup B,A\cap B=\emptyset,\forall uv\in E:u\in A\oplus v\in A$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $K_{m,n}$
+\end_inset
+
+ je poln dvodelni graf, 
+\begin_inset Formula $\left|A\right|=m$
+\end_inset
+
+, 
+\begin_inset Formula $\left|B\right|=n$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ dvodelen 
+\begin_inset Formula $\Leftrightarrow\forall$
+\end_inset
+
+ komponenta 
+\begin_inset Formula $G$
+\end_inset
+
+ dvodelna
+\end_layout
+
+\begin_layout Standard
+Pot, sod cikel, hiperkocka so dvodelni, petersenov ni.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ dvodelen 
+\begin_inset Formula $\Leftrightarrow G$
+\end_inset
+
+ ne vsebuje lihega cikla.
+\end_layout
+
+\begin_layout Standard
+Biparticija glede na parnost 
+\begin_inset Formula $d_{G}\left(u,x_{0}\right)$
+\end_inset
+
+, 
+\begin_inset Formula $x_{0}$
+\end_inset
+
+ fiksen.
+\end_layout
+
+\begin_layout Standard
+Dvodelen 
+\begin_inset Formula $k-$
+\end_inset
+
+regularen, 
+\begin_inset Formula $\left|E\right|\ge1\Rightarrow$
+\end_inset
+
+
+\begin_inset Formula $\left|A\right|=\left|B\right|$
+\end_inset
+
+.
+ Dokaz: 
+\begin_inset Formula $\sum_{u\in A}\deg u=\left|E\right|=\cancel{k}\left|A\right|=\cancel{k}\left|B\right|=\sum_{u\in B}\deg u$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Krožni
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\Cir\left(n,\left\{ s_{1},\dots,s_{k}\right\} \right):\left|V\right|=n,$
+\end_inset
+
+ množica preskokov
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\Omega\Cir\left(n,\left\{ s,n-s\right\} \right)=\gcd\left\{ n,s\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $W_{n}$
+\end_inset
+
+ (kolo) pa je cikel z univerzalnim vozliščem.
+\end_layout
+
+\begin_layout Paragraph
+Homomorfizem
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\varphi:VG\to VH$
+\end_inset
+
+ slika povezave v povezave
+\end_layout
+
+\begin_layout Standard
+Primer: 
+\begin_inset Formula $K_{2}$
+\end_inset
+
+ je homomorfna slika 
+\begin_inset Formula $\forall$
+\end_inset
+
+ bipartitnega grafa.
+\end_layout
+
+\begin_layout Standard
+V povezavah in vozliščih surjektiven 
+\begin_inset Formula $hm\varphi$
+\end_inset
+
+ je epimorfizem.
+\end_layout
+
+\begin_layout Standard
+V vozliščih injektiven 
+\begin_inset Formula $hm\varphi$
+\end_inset
+
+ je monomorfizem/vložitev.
+\end_layout
+
+\begin_layout Standard
+Vložitev, ki ohranja razdalje, je izometrična.
+\end_layout
+
+\begin_layout Standard
+Kompozitum homomorfizmov je spet homomorfizem.
+\end_layout
+
+\begin_layout Standard
+Izomorfizem je bijektivni 
+\begin_inset Formula $hm\varphi$
+\end_inset
+
+ s homomorfnim inverzom.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $im\varphi$
+\end_inset
+
+ 
+\begin_inset Formula $f:VG\to VH$
+\end_inset
+
+ 
+\begin_inset Formula $\forall u,v\in VG:uv\in EG\Leftrightarrow fufv\in EH$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Nad množico vseh grafov 
+\begin_inset Formula $\mathcal{G}$
+\end_inset
+
+ je izomorfizem (
+\begin_inset Formula $\cong$
+\end_inset
+
+) ekv.
+ rel.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $im\varphi$
+\end_inset
+
+ 
+\begin_inset Formula $G\to G$
+\end_inset
+
+ je avtomorfizem.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\Aut G$
+\end_inset
+
+ je grupa avtomorfizmov s komponiranjem.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\Aut K_{n}=\left\{ \pi\in S_{n}=\text{permutacije }n\text{ elementov}\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\Aut P_{n}=\left\{ \text{trivialni }id,f\left(i\right)=n-i-1\right\} $
+\end_inset
+
+, 
+\begin_inset Formula $\Aut G\cong\Aut\overline{G}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Izomorfizem ohranja stopnje, št.
+ 
+\begin_inset Formula $C_{4}$
+\end_inset
+
+, povezanost, 
+\begin_inset Formula $\left|EG\right|,\dots$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G\cong\overline{G}\Rightarrow\left|VG\right|\%4\in\left\{ 0,1\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Operacije
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $H$
+\end_inset
+
+ vpeti podgraf 
+\begin_inset Formula $G\Leftrightarrow\exists F\subseteq EG\ni:H=G-F$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $H$
+\end_inset
+
+ inducirani podgraf 
+\begin_inset Formula $G\Leftrightarrow\exists S\subseteq VG\ni:H=G-S$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $H$
+\end_inset
+
+ podgraf 
+\begin_inset Formula $G\Leftrightarrow\exists S\subseteq VG,F\subseteq EG\ni:H=\left(G-F\right)-S$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $uv=e\in EG$
+\end_inset
+
+.
+ 
+\begin_inset Formula $G/e$
+\end_inset
+
+ je skrčitev.
+ (identificiramo 
+\begin_inset Formula $u=v$
+\end_inset
+
+)
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $H$
+\end_inset
+
+ minor 
+\begin_inset Formula $G$
+\end_inset
+
+: 
+\begin_inset Formula $H=f_{1}...f_{k}G$
+\end_inset
+
+ za 
+\begin_inset Formula $f_{i}$
+\end_inset
+
+ skrčitev/odstranjevanje
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $VG^{+}e\coloneqq VG\cup\left\{ x_{e}\right\} $
+\end_inset
+
+, 
+\begin_inset Formula $EG^{+}e\coloneqq EG\setminus e\cup\left\{ x_{e}u,x_{e}v\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $VG^{+}e$
+\end_inset
+
+ je subdivizija, 
+\begin_inset Formula $e\in EG$
+\end_inset
+
+.
+ Na 
+\begin_inset Formula $e$
+\end_inset
+
+ dodamo vozlišče.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $H$
+\end_inset
+
+ subdivizija 
+\begin_inset Formula $G$
+\end_inset
+
+ 
+\begin_inset Formula $\Leftrightarrow H=G^{+}\left\{ e_{1},\dots,e_{k}\right\} ^{+}\left\{ f_{1}\dots f_{j}\right\} ^{+}\dots$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Stopnja vozlišč se s subdivizijo ne poveča.
+\end_layout
+
+\begin_layout Standard
+Glajenje 
+\begin_inset Formula $G^{-}v$
+\end_inset
+
+, 
+\begin_inset Formula $v\in VG$
+\end_inset
+
+ je obrat subdivizije.
+ 
+\begin_inset Formula $\deg_{G}v=2$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ in 
+\begin_inset Formula $H$
+\end_inset
+
+ sta homeomorfna, če sta subdivizija istega grafa.
+\end_layout
+
+\begin_layout Standard
+Kartezični produkt: 
+\begin_inset Formula $V\left(G\square H\right)\coloneqq VG\times VH$
+\end_inset
+
+, 
+\begin_inset Formula $E\left(G\square H\right)\coloneqq\left\{ \left\{ \left(g,h\right),\left(g',h'\right)\right\} ;g=g'\wedge hh'\in EH\vee h=h'\wedge gg'\in EG\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left(\mathcal{G},\square\right)$
+\end_inset
+
+ monoid, enota 
+\begin_inset Formula $K_{1}$
+\end_inset
+
+.
+ 
+\begin_inset Formula $Q_{n}\cong Q_{n-1}\square K_{2}=Q_{n-2}\square K_{2}^{\square,2}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Disjunktna unija: 
+\begin_inset Formula $G$
+\end_inset
+
+, 
+\begin_inset Formula $H$
+\end_inset
+
+ disjunktna.
+ 
+\begin_inset Formula $V\left(G\cup H\right)\coloneqq VG\cup VH$
+\end_inset
+
+, 
+\begin_inset Formula $E\left(G\cup H\right)\coloneqq EG\cup EH$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G,H$
+\end_inset
+
+ dvodelna 
+\begin_inset Formula $\Rightarrow G\square H$
+\end_inset
+
+ dvodelen
+\end_layout
+
+\begin_layout Paragraph
+\begin_inset Formula $k-$
+\end_inset
+
+povezan graf
+\end_layout
+
+\begin_layout Standard
+ima 
+\begin_inset Formula $\geq k+1$
+\end_inset
+
+ vozlišč in ne vsebuje prerezne množice moči 
+\begin_inset Formula $<k$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $X\subseteq VG$
+\end_inset
+
+ je prerezna množica 
+\begin_inset Formula $\Leftrightarrow\Omega\left(G-X\right)>\Omega G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $Y\subseteq EG$
+\end_inset
+
+ prerezna množica povezav 
+\begin_inset Formula $\Leftrightarrow\Omega\left(G-Y\right)>\Omega G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Povezanost grafa: 
+\begin_inset Formula $\kappa G=\max k$
+\end_inset
+
+, da je 
+\begin_inset Formula $G$
+\end_inset
+
+ 
+\begin_inset Formula $k-$
+\end_inset
+
+povezan.
+ Primeri: 
+\begin_inset Formula $\kappa K_{n}=n-1$
+\end_inset
+
+, 
+\begin_inset Formula $\kappa P_{n}=1$
+\end_inset
+
+, 
+\begin_inset Formula $\kappa C_{n}=2$
+\end_inset
+
+, 
+\begin_inset Formula $\kappa K_{n,m}=\min\left\{ n,m\right\} $
+\end_inset
+
+, 
+\begin_inset Formula $\kappa Q_{n}=n$
+\end_inset
+
+, 
+\begin_inset Formula $\kappa G\text{\ensuremath{\leq}}\delta G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Izrek (Menger): 
+\begin_inset Formula $\left|VG\right|\geq k+1$
+\end_inset
+
+: 
+\begin_inset Formula $G$
+\end_inset
+
+ 
+\begin_inset Formula $k-$
+\end_inset
+
+povezan 
+\begin_inset Formula $\Leftrightarrow\forall u,v\in VG,uv\not\in EG:\exists k$
+\end_inset
+
+ notranje disjunktnih 
+\begin_inset Formula $u,v-$
+\end_inset
+
+poti
+\end_layout
+
+\begin_layout Standard
+Graf je 
+\begin_inset Formula $k-$
+\end_inset
+
+povezan po povezavah, če ne vsebuje prerezne množice povezav moči 
+\begin_inset Formula $<k$
+\end_inset
+
+.
+ Povezanost grafa po povezavah: 
+\begin_inset Formula $\kappa'G=\max k$
+\end_inset
+
+, da je 
+\begin_inset Formula $G$
+\end_inset
+
+ 
+\begin_inset Formula $k-$
+\end_inset
+
+povezan po povezavah.
+ Primeri: 
+\begin_inset Formula $\kappa'K_{n}=n-1$
+\end_inset
+
+, 
+\begin_inset Formula $\kappa'P_{n}=1$
+\end_inset
+
+, 
+\begin_inset Formula $\kappa'C_{n}=2$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Izrek (Menger'): 
+\begin_inset Formula $G$
+\end_inset
+
+ 
+\begin_inset Formula $k-$
+\end_inset
+
+povezan 
+\begin_inset Formula $\Leftrightarrow\forall u,v\in VG\exists k$
+\end_inset
+
+ po povezavah disjunktnih 
+\begin_inset Formula $u,v-$
+\end_inset
+
+poti
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\forall G\in\mathcal{G}:\kappa G\leq\kappa'G\leq\delta G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Drevo
+\end_layout
+
+\begin_layout Standard
+je povezan gozd.
+ Gozd je graf brez ciklov.
+\end_layout
+
+\begin_layout Standard
+Drevo z vsaj dvema vozliščema premore dva lista.
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+hspace*{
+\backslash
+fill}
+\end_layout
+
+\end_inset
+
+NTSE:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ drevo 
+\begin_inset Formula $\Leftrightarrow\forall u,v\in VG\exists!u,v-$
+\end_inset
+
+pot 
+\begin_inset Formula $\Leftrightarrow\Omega G=1\wedge\forall e\in EG:e$
+\end_inset
+
+ most 
+\begin_inset Formula $\Leftrightarrow\Omega G=1\wedge\left|EG\right|=\left|VG\right|-1$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Vpeto drevo grafa je vpet podgraf, ki je drevo.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\tau G\sim$
+\end_inset
+
+ število vpetih dreves.
+ 
+\begin_inset Formula $\Omega G=1\Leftrightarrow\tau G\geq1$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\forall e\in EG:\tau G=\tau\left(G-e\right)+\tau\left(G/e\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\text{\text{\text{Laplaceova matrika: }}}L\left(G\right)_{ij}=\begin{cases}
+\deg_{G}v_{i} & ;i=j\\
+-\left(\text{št. uv povezav}\right) & ;\text{drugače}
+\end{cases}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Izrek (Kirchoff): za 
+\begin_inset Formula $G$
+\end_inset
+
+ povezan multigraf je 
+\begin_inset Formula $\forall i:\tau G=\det\left(LG\text{ brez \ensuremath{i}te vrstice in \ensuremath{i}tega stolpca}\right)$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\tau K_{n}=n^{n-2}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Prüferjeva koda, če lahko linearno uredimo vozlišča: Ponavljaj, dokler 
+\begin_inset Formula $VG\not=\emptyset$
+\end_inset
+
+: vzemi prvi list, ga odstrani in v vektor dodaj njegovega soseda.
+\end_layout
+
+\begin_layout Standard
+Blok je maksimalen povezan podgraf brez prereznih vozlišč.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\tau G=\tau B_{1}\cdot\cdots\cdot\tau B_{k}$
+\end_inset
+
+ za bloke 
+\begin_inset Formula $\vec{B}$
+\end_inset
+
+ grafa 
+\begin_inset Formula $G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{multicols}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{multicols}{2}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Eulerjev
+\end_layout
+
+\begin_layout Standard
+sprehod v m.grafu vsebuje vse povezave po enkrat.
+\end_layout
+
+\begin_layout Standard
+Eulerjev obhod je sklenjen eulerjev sprehod.
+\end_layout
+
+\begin_layout Standard
+Eulerjev graf premore eulerjev obhod.
+\end_layout
+
+\begin_layout Standard
+Za povezan multigraf 
+\begin_inset Formula $G$
+\end_inset
+
+ eulerjev 
+\begin_inset Formula $\Leftrightarrow\forall v\in VG:\deg_{G}v$
+\end_inset
+
+ sod
+\end_layout
+
+\begin_layout Standard
+Fleuryjev algoritem za eulerjev obhod v eulerjevem grafu: Začnemo v poljubni
+ povezavi, jo izbrišemo, nadaljujemo na mostu le, če nimamo druge možne
+ povezave.
+\end_layout
+
+\begin_layout Standard
+Dekompozicija: delitev na povezavno disjunktne podgrafe.
+\end_layout
+
+\begin_layout Standard
+Dekompozicija je lepa, če so podgrafi izomorfni.
+ (
+\begin_inset Formula $\exists$
+\end_inset
+
+ za 
+\begin_inset Formula $P_{5,2}$
+\end_inset
+
+)
+\end_layout
+
+\begin_layout Standard
+Vsak eulerjev graf premore dekompozicijo v cikle.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $Q_{n}$
+\end_inset
+
+ eulerjev 
+\begin_inset Formula $\Leftrightarrow n$
+\end_inset
+
+ sod
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $K_{m,n,p}$
+\end_inset
+
+ eulerjev 
+\begin_inset Formula $\Leftrightarrow m,n,p$
+\end_inset
+
+ iste parnosti
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ eulerjev 
+\begin_inset Formula $\wedge H$
+\end_inset
+
+ eulerjev 
+\begin_inset Formula $\Rightarrow G\square H$
+\end_inset
+
+ eulerjev
+\end_layout
+
+\begin_layout Paragraph
+Hamiltonov
+\end_layout
+
+\begin_layout Standard
+cikel vsebuje vsa vozlišča grafa.
+\end_layout
+
+\begin_layout Standard
+Hamilton graf premore Hamiltonov cikel.
+\end_layout
+
+\begin_layout Standard
+Hamiltonova pot vsebuje vsa vozlišča.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ hamiltonov, 
+\begin_inset Formula $S\subseteq VG\Rightarrow\Omega\left(G-S\right)\leq\left|S\right|$
+\end_inset
+
+ torej:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\exists S\in VG:\Omega\left(G-S\right)>\left|S\right|\Rightarrow G$
+\end_inset
+
+ ni hamiltonov.
+ Primer: 
+\begin_inset Formula $G$
+\end_inset
+
+ vsebuje prerezno vozlišče 
+\begin_inset Formula $\Rightarrow G$
+\end_inset
+
+ ni hamiltonov.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $K_{n,m}$
+\end_inset
+
+ je hamiltonov 
+\begin_inset Formula $\Leftrightarrow m=n$
+\end_inset
+
+ (za 
+\begin_inset Formula $S$
+\end_inset
+
+ vzamemo 
+\begin_inset Formula $\min\left\{ m,n\right\} $
+\end_inset
+
+)
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left|VG\right|\geq3,\forall u,v\in VG:\deg_{G}u+\deg_{G}v\geq\left|VG\right|\Rightarrow G$
+\end_inset
+
+ hamil.
+\end_layout
+
+\begin_layout Standard
+Dirac: 
+\begin_inset Formula $\left|VG\right|\geq3,\forall u\in VG:\deg_{G}u\geq\frac{\left|VG\right|}{2}\Rightarrow G$
+\end_inset
+
+ hamilton.
+\end_layout
+
+\begin_layout Paragraph
+Ravninski
+\end_layout
+
+\begin_layout Standard
+graf brez sekanja povezav narišemo v ravnino
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $K_{2,3}$
+\end_inset
+
+ je ravninski, 
+\begin_inset Formula $K_{3,3}$
+\end_inset
+
+, 
+\begin_inset Formula $K_{5}$
+\end_inset
+
+, 
+\begin_inset Formula $C_{5}\square C_{5}$
+\end_inset
+
+ in 
+\begin_inset Formula $P_{5,2}$
+\end_inset
+
+ niso ravninski.
+\end_layout
+
+\begin_layout Standard
+Vložitev je ravninski graf z ustrezno risbo v ravnini.
+\end_layout
+
+\begin_layout Standard
+Lica so sklenjena območja vložitve brez vozlišč in povezav.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ lahko vložimo v ravnino 
+\begin_inset Formula $\Leftrightarrow G$
+\end_inset
+
+ lahko vložimo na sfero.
+\end_layout
+
+\begin_layout Standard
+Dolžina lica 
+\begin_inset Formula $F\sim lF$
+\end_inset
+
+ je št.
+ povezav obhoda lica.
+\end_layout
+
+\begin_layout Standard
+Drevo je ravninski graf z enim licem dolžine 
+\begin_inset Formula $2\left|ET\right|$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\sum_{F\in FG}lF=2\left|EG\right|$
+\end_inset
+
+, 
+\begin_inset Formula $lF\geq gG$
+\end_inset
+
+ za ravninski 
+\begin_inset Formula $G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $2\left|EG\right|=\sum_{F\in FG}lF\geq\sum_{F\in FG}gG=gG\left|FG\right|$
+\end_inset
+
+ (
+\begin_inset Formula $G$
+\end_inset
+
+ ravn.)
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ ravninski 
+\begin_inset Formula $\Rightarrow\left|EG\right|\geq\frac{gG}{2}$
+\end_inset
+
+
+\begin_inset Formula $\left|FG\right|$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Euler: 
+\begin_inset Formula $\left|VG\right|-\left|EG\right|+\left|FG\right|=1+\Omega G$
+\end_inset
+
+ za ravninski 
+\begin_inset Formula $G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ ravninski, 
+\begin_inset Formula $\left|VG\right|\geq3\Rightarrow\left|EG\right|\leq3\left|VG\right|-6$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ ravninski brez 
+\begin_inset Formula $C_{3}$
+\end_inset
+
+, 
+\begin_inset Formula $\left|VG\right|\geq3\Rightarrow\left|EG\right|\text{\ensuremath{\leq2\left|VG\right|-4}}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Triangulacija je taka vložitev, da so vsa lica omejena s 
+\begin_inset Formula $C_{3}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+V maksimalen ravninski graf ne moremo dodati povezave, da bi ostal ravninski.
+ 
+\begin_inset Formula $\sim$
+\end_inset
+
+ Ni pravi vpet podgraf ravn.
+ grafa.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $K_{5}-e$
+\end_inset
+
+ je maksimalen ravninski graf 
+\begin_inset Formula $\forall e\in EK_{5}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ ravninski.
+ NTSE: 
+\begin_inset Formula $G$
+\end_inset
+
+ triangulacija 
+\begin_inset Formula $\Leftrightarrow G$
+\end_inset
+
+ maksimalen ravninski 
+\begin_inset Formula $\Leftrightarrow\left|EG\right|=3\left|VG\right|-6$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ ravninski 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ vsaka subdivizija 
+\begin_inset Formula $G$
+\end_inset
+
+ ravninska.
+\end_layout
+
+\begin_layout Standard
+Kuratovski: 
+\begin_inset Formula $G$
+\end_inset
+
+ ravn.
+ 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ ne vsebuje subdivizije 
+\begin_inset Formula $K_{5}$
+\end_inset
+
+ ali 
+\begin_inset Formula $K_{3,3}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Wagner: 
+\begin_inset Formula $G$
+\end_inset
+
+ ravn.
+ 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ niti 
+\begin_inset Formula $K_{5}$
+\end_inset
+
+ niti 
+\begin_inset Formula $K_{3,3}$
+\end_inset
+
+ nista njegova minorja
+\end_layout
+
+\begin_layout Standard
+Zunanje-ravninski ima vsa vozlišča na robu zunanjega lica.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $2-$
+\end_inset
+
+povezan zunanje-ravninski je 
+\series bold
+hamiltonov
+\series default
+.
+\end_layout
+
+\begin_layout Standard
+Zunanje-ravninski 
+\begin_inset Formula $G$
+\end_inset
+
+, 
+\begin_inset Formula $\left|VG\right|\geq2$
+\end_inset
+
+ ima vozlišče stopnje 
+\begin_inset Formula $\leq2$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Paragraph
+Barvanje
+\end_layout
+
+\begin_layout Standard
+je taka 
+\begin_inset Formula $C:VG\to\left\{ 1..k\right\} \Leftrightarrow\forall uv\in EG:Cu\not=Cv$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Kromatično število 
+\begin_inset Formula $\chi G$
+\end_inset
+
+ je najmanjši 
+\begin_inset Formula $k$
+\end_inset
+
+, 
+\begin_inset Formula $\ni:\exists$
+\end_inset
+
+ 
+\begin_inset Formula $k-$
+\end_inset
+
+barvanje 
+\begin_inset Formula $G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Primer: 
+\begin_inset Formula $\chi K_{n}=n$
+\end_inset
+
+, 
+\begin_inset Formula $\chi C_{n}=\begin{cases}
+2 & ;n\text{ sod}\\
+3 & ;n\text{ lih}
+\end{cases}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{multicols}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document