1 23456789 0 :;4 01 <4=5<=42 00 78 08 2<>;4? 01 572852 0 @ A374 0B-01 =AA7C38 0 !"#$%$&$'()*+(,*-.&$/$%*"/0(/"1(23.'+%-'"$/0(4/$5'-/$%6#(7!,2849( :*+;#5*.(*"(4/$5'-/$%6#(%"(</$/(=6%'"6'( >&0?(@AB(CADE( -./0123*4//1506) !"#$%&'(""$)#*+*,'(-./(0 !"#$%&'()*!+',!') 0 http://www.csc.ncsu.edu/faculty/bdsullivan
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<\.0G%$,S00 • ! `&+(G+/.5(%#"\$(Q&#$( /"?(*01(G+/.5a( • ! 4.#"b[M%"/0*.G[M%S'0MZG.*0\#).0 6*--&"%$?(#$+&6$&+' I0 G\.Qa%.0E"M,#""Q0/.*'.H0#*/0EF"M!#""Q0'R#%'.H00 14
='.Z("01R#%'$GQ0h)-0 !"#$ I0<%..01G%(,G(%.0 58',*,9&#'!":&#'31$+.#'&"#*? )) • ! V 3j >71I0:$*/0G\.0 -/3%-&-(['%G5$'1(%"1'.'"1'"$(#'$ 0$*0 P( (17,15) U0 (8,10) (7,5) O0 p0 (3,6) (4,1) L0 T0 L0 U0 (3,0) (2,0) ( L0 U0 O0 O0 ( (3,0) (2,0) (1,0) (1,0) &F1:),$MF027)E2<O/8G)F0:)0)/16802)0/=<21>FG)<6)>288:P) 15 [Some prefer to think of belief propagation -- also efficient (and exact) on trees]
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� � � � 1R#%'.09%#R\06$.%#%,\Q0 Nowhere dense r Locally bounded r Locally excluding Bounded expansion expansion a minor Excluding a topological minor r Locally bounded Excluding a minor treewidth Bounded genus Planar Bounded treewidth Bounded degree Outerplanar Bounded treedepth Forests Star forests Linear forests 17
:k<03"FM%$G\J'0q09%#R\01G%(,G(%.I0k%M'0q05M*'0 H*$# 0MZ0R%M!".J'0!.,MJ.0:k<I0 0 • ! 1 <27824 0< 422 0h!M(*/./0/.F.*.%#,Qi0 • ! 1 =@943k6 7 1;V;4k671V 0h!M(*/./0.fR#*'$M*i0 • ! B ;V783<789 1 2< 0h!M(*/./0F.*('i0 • ! V 3j >710h!M(*/./0G%..[$/G\i00 0 3*/0G\.%.0#%.0J.G#bG\.M%.J'`00 • ! :;bJM/."0,\.,S$*F0 *"("*[5'+'K1'"#'(G+/.5#b 0h ; 0r0ZM%J("#0'$l.i00 • ! 2V1;bJM/."0,\.,S$*F0 ./+/-'$'+%X'1(F?($+''[%1$5 ' @AB' ( C&"97D8/9)'+&3D8/:#' -%G5$("*$()/00(%"$*(/"?(*)($5'#'(6/$'G*+%'#Z(( :*+#'B(%$\#( 1"/)'38'3' -'-F'+#5%.(7]('3%#$%"G(+'#&0$#(/+'("'G/$%N'9( T5'(/0G*+%$5-#(*)$'"(5/N'(c5&G'd( 1$))&+',8+#3"+3#'' 18 *Recall, this was broadest class. And lots of problems are expressible in FO-logic.
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1R#%'$GQ0U-XI0@M(*/./02fR#*'$M*0 • ! 30F%#R\0,"#''0 P 0\#'0 F*&"1'1('3./"#%*"( $Z0.).%Q0 + b'\#""M[0 hGMRM"MF$,#"i0J$*M%0\#'0/.*'$GQ0#G0JM'G0 )7+9 -00 | E ( H ) | O r ( G ) := sup O r ( G ) ≤ f ( r ) O r ( G ) = max | V ( H ) | H ∈ G O r 0 G ∈ G 0 0 0 0 0 ,<>8H) %."#G./0:k<0#"FM%$G\J'0 1*"\$ 0%.g($%.0S*M[$*F0 )7+9 ( 22
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