Added some pictures for single edge and two path patterns.

macros.tex

@@ -106,9 +106,25 @@
 \newcommand{\out}{output}%output aggregation over the output vector
 \newcommand{\numocc}[2]{\#\left(#1, #2\right)}
 	%Graph Symbols
-	\newcommand{\ed}{|}
+	\newcommand{\ed}{
+					\begin{tikzpicture}%[baseline=0.00005cm]
+					%\begin{scope}[yshift=-0.1cm]
+						\node at (0, 0)[fill, draw, circle, inner sep=0pt, minimum size=2pt](bottom){};
+						\node [above=0.07cm of bottom, fill, draw, circle, inner sep=0pt, minimum size=2pt](top){};
+						\draw[semithick] (top) -- (bottom);
+					%\end{scope}
+					\end{tikzpicture}
+				}
 	\newcommand{\twodis}{\|}
-	\newcommand{\twopath}{\land}
+	\newcommand{\twopath}{
+						\begin{tikzpicture}[every node/.style={circle, draw, fill, inner sep=0pt, minimum size=2pt}]
+							\node at (0, 0.08) (top){};
+							\node [below left=0.08cm and 0.01cm of top](left){};
+							\node[below right=0.08cm and 0.01cm of top](right){};
+							\draw[semithick](top) -- (left);
+							\draw[semithick](top) -- (right);
+						\end{tikzpicture}
+					}
 	\newcommand{\threedis}{| | |}
 	\newcommand{\tri}{\triangle}
 	\newcommand{\twopathdis}{| \land}

main.tex

@@ -3,6 +3,7 @@
 
 \usepackage{algpseudocode}
 \usepackage{algorithm}
+\usepackage{tikz}
 
 \usepackage{comment}
 \usepackage{amsmath}

poly-form.tex

@@ -110,7 +110,7 @@ To this end, consider the following graph $G(V, E)$, where $|E| = \numedge$, $|V
 Before proceeding, let us list all possible edge patterns in an arbitrary $G$ consisting of $\leq 3$ distinct edges.
 
 \begin{itemize}
-	\item Single Edge ($\ed$)
+	\item Single Edge $\left(\ed\right)$
 	\item 2-path ($\twopath$)
 	\item 2-matching ($\twodis$)
 	\item Triangle ($\tri$)

ra-to-poly.tex

@@ -2,6 +2,27 @@
 %!TEX root=./main.tex
 \onecolumn
 \section{Query translation into polynomials}
+
+\begin{tikzpicture}[every node/.style={circle, draw=black, fill=black, text=white}]
+	\node{1}
+		child { node{2}
+			child { node{4}}
+			child { node{5}}
+		}
+		child{ node{3} };
+     \node at (5, 5) [fill,circle,inner sep=0pt,minimum size=3pt] (top) {};
+     \node[fill,circle,inner sep=0pt,minimum size=3pt, below=0.5cm of top] (bottom){};
+	\draw (top)--(bottom);
+%	\draw[red, very thick] (0, 0) rectangle (4, 4);
+%	\draw[blue, very thin] (0, 0) circle (2cm);
+%	\draw (0, 0) ellipse (4cm and 2cm);
+%	\draw (4, 4) arc (0:180:4);
+%	\fill[olive] (0, 0) rectangle (3, 3);
+%	\fill[teal] (-2, 2) circle (1cm);
+%	\fill[blue] (1, 1) circle(2pt);
+%	\fill[blue!50!red](3, 3) circle(.2cm);
+%	\fill[blue!50] (4, 4) circle (3pt);
+\end{tikzpicture}
 %\AH{This section will involve the set of queries (RA+) that we are interested in, the probabilistic/incomplete models we address, and the outer aggregate functions we perform over the output \textit{annotation}
 %1) RA notation
 %2) DB (TIDB) notation