\n{$C_k$}{cycle on~$k$ vertices}
\n{${\cal D}(G)$}{optimal MSF decision tree for a~graph~$G$ \[decdef]}
\n{$D(G)$}{depth of ${\cal D}(G)$ \[decdef]}
-\n{$D(m,n)$}{decision tree complexity of MSF \[decdef]}
+\n{$D(m,n)$}{decision tree complexity of MSF for $m$~edges and $n$~vertices \[decdef]}
\n{$D_n$}{$n\times n$ matrix with 0's on the main diagonal and 1's elsewhere \[hatrank]}
\n{$\deg_G(v)$}{degree of vertex~$v$ in graph~$G$; we omit $G$ if it is clear from context}
\n{$E(G)$}{set of edges of a graph~$G$}
\n{$K_k$}{complete graph on~$k$ vertices}
\n{$L(\pi,A)$}{lexicographic ranking function for permutations on a~set~$A\subseteq{\bb N}$ \[brackets]}
\n{$L^{-1}(i,A)$}{lexicographic unranking function, the inverse of~$L$ \[brackets]}
-\n{$\log n$}{a binary logarithm of the number~$n$}
+\n{$\log n$}{binary logarithm of the number~$n$}
\n{$\log^* n$}{iterated logarithm: $\log^*n := \min\{i \mid \log^{(i)}n \le 1\}$; the inverse of~$2\tower n$}
\n{$\<LSB>(x)$}{position of the lowest bit set in~$x$ \[lsbmsb]}
\n{$\<MSB>(x)$}{position of the highest bit set in~$x$ \[lsbmsb]}
\n{$\mst(G)$}{the unique minimum spanning tree of a graph~$G$ \[mstnota]}
\n{$m(G)$}{number of edges of a graph~$G$, that is $\vert E(G)\vert$}
\n{$m$}{$m(G)$ when the graph~$G$ is clear from context}
-\n{$\bb N$}{set of all natural numbers, including 0}
+\n{$\bb N$}{set of all non-negative integers}
\n{${\bb N}^+$}{set of all positive integers}
\n{$N_0(M)$}{number of permutations satisfying the restrictions~$M$ \[restnota]}
\n{$n(G)$}{number of vertices of a graph~$G$, that is $\vert V(G)\vert$}
\n{$X \choose k$}{the set of all $k$-element subsets of a set~$X$}
\n{$G/e$}{multigraph contraction \[contract]}
\n{$G\sgc e$}{simple graph contraction \[simpcont]}
-\n{$G/X$, $G.X$}{contraction by a~set $X$ of vertices or edges \[setcont]}
+\n{$G/X$, $G\sgc X$}{contraction by a~set $X$ of vertices or edges \[setcont]}
\n{$f[X]$}{function applied to a set: $f[X]:=\{ f(x) \mid x\in X \}$}
\n{$f[e]$}{as edges are two-element sets, $f[e]$ maps both endpoints of an edge~$e$}
\n{$f^{(i)}$}{function~$f$ iterated $i$~times: $f^{(0)}(x):=x$, $f^{(i+1)}(x):=f(f^{(i)}(x))$}
\n{$\sigma^k$}{the string~$\sigma$ repeated $k$~times \[bitnota]}
\n{$\0$, $\1$}{bits in a~bit string \[bitnota]}
\n{$\equiv$}{congruence modulo a~given number}
-\n{$\bf x$}{a~vector with elements $x_1,\ldots,x_d$; $x$ is its bitwise encoding \[vecnota]}
+\n{$\bf x$}{vector with elements $x_1,\ldots,x_d$; $x$ is its bitwise encoding \[vecnota]}
\n{$x \shl n$}{bitwise shift of~$x$ by $n$~positions to the left: $x\shl n = x\cdot 2^n$}
\n{$x \shr n$}{bitwise shift of~$x$ by $n$~positions to the right: $x\shr n = \lfloor x/2^n \rfloor$}
\n{$[n]$}{the set $\{1,2,\ldots,n\}$ \[pranksect]}
-\n{$n^{\underline k}$}{the $k$-th falling factorial power: $n\cdot(n-1)\cdot\ldots\cdot(n-k+1)$ \[kpranksect]}
+\n{$n^{\underline k}$}{$k$-th falling factorial power: $n\cdot(n-1)\cdot\ldots\cdot(n-k+1)$ \[kpranksect]}
\n{$H\minorof G$}{$H$ is a~minor of~$G$ \[minordef]}
\n{$G\crpt R$}{graph~$G$ with edges in~$R$ corrupted \[corrnota]}
\n{$R^C$}{$R^C = R\cap \delta(C)$ \[corrnota]}
(if the edge is a loop).
\proclaim{Notation}%
-When the meaning is clear from the context, we use our notation originally
-defined for graphs even for multigraphs. For example, $xy\in E(G)$ becomes a
+When the meaning is clear from the context, we use the standard graph notation
+even for multigraphs. For example, $xy\in E(G)$ becomes a
shorthand for $\exists e\in E(G)$ such that $M(G)(e) = \{x,y\}$. Also, we
consider multigraphs with no multiple edges nor loops and simple graphs to be
the same objects, although they formally differ.
$$\eqalign{
V' &= (V(G) \setminus \{x,y\}) \cup \{v_e\},\quad\hbox{where $v_e$ is a new vertex,}\cr
E' &= E(G) - \{e\},\cr
-M'(f) &= \{ m(v) \mid v\in M(f) \} \quad\hbox{for every $f=\in E'$, and}\cr
-m(x) &= \cases{v_e & \hbox{for $v=x,y,$}\cr v & \hbox{otherwise.}} \cr
+M'(f) &= \{ m(v) \mid v\in M(f) \} \quad\hbox{for every $f\in E'$, and}\cr
+m(x) &= \cases{v_e & \hbox{for $v=x,y,$}\cr \noalign{\vskip5pt} v & \hbox{otherwise.}} \cr
}$$
-Sometimes we need contraction for simple graphs as well. It is equivalent to performing
+We sometimes also need to contract edges in simple graphs. It is equivalent to performing
the multigraph contraction and then unifying parallel edges and deleting loops.
\defn\id{simpcont}%
\defn\id{setcont}%
We can also extend the above definitions to contractions of a~set of vertices or edges.
For $F\subseteq E(G)$, the graph $G/F$ is defined as $(G/f_1)/f_2/\ldots/f_k$ where
-$f_1,\ldots,f_k$ are the elements of~$F$ (you can observe that the result
-does not depend on the order of edges). For $U\subseteq V(G)$, we define $G/U$
+$f_1,\ldots,f_k$ are the elements of~$F$ (the result obviously does not depend on the order of edges).
+For $U\subseteq V(G)$, we define $G/U$
as the graph with all vertices of~$U$ merged to a~single vertex, that is $(G\cup U^*)/U^*$,
where $U^*$ is the complete graph on~$U$. Similarly for $G\sgc F$ and $G\sgc U$.
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