diff --git a/pretext/Graphs/AnAdjacencyList.ptx b/pretext/Graphs/AnAdjacencyList.ptx
index 154fb979..e30119be 100755
--- a/pretext/Graphs/AnAdjacencyList.ptx
+++ b/pretext/Graphs/AnAdjacencyList.ptx
@@ -19,5 +19,8 @@
to compactly represent a sparse graph. The adjacency list also allows us
to easily find all the links that are directly connected to a particular
vertex.
+
+
+
diff --git a/pretext/Graphs/AnAdjacencyMatrix.ptx b/pretext/Graphs/AnAdjacencyMatrix.ptx
index 37022858..1a0e7197 100755
--- a/pretext/Graphs/AnAdjacencyMatrix.ptx
+++ b/pretext/Graphs/AnAdjacencyMatrix.ptx
@@ -92,5 +92,8 @@
is connected to every other vertex. There are few real problems that
approach this sort of connectivity. The problems we will look at in this
chapter all involve graphs that are sparsely connected.
+
+
+
diff --git a/pretext/Graphs/AnalysisofDijkstrasAlgorithm.ptx b/pretext/Graphs/AnalysisofDijkstrasAlgorithm.ptx
index a5403684..957173cf 100755
--- a/pretext/Graphs/AnalysisofDijkstrasAlgorithm.ptx
+++ b/pretext/Graphs/AnalysisofDijkstrasAlgorithm.ptx
@@ -11,5 +11,8 @@
for loop is executed once for each edge in the
graph, and within the for loop the call to decreaseKey takes
time O(E\log(V)). So the combined running time is O((V+E) \log(V)).
+
+
+
diff --git a/pretext/Graphs/BreadthFirstSearchAnalysis.ptx b/pretext/Graphs/BreadthFirstSearchAnalysis.ptx
index ac87520b..ed3aedba 100755
--- a/pretext/Graphs/BreadthFirstSearchAnalysis.ptx
+++ b/pretext/Graphs/BreadthFirstSearchAnalysis.ptx
@@ -20,5 +20,8 @@
Finally, at least for this problem, there is the time required to build
the initial graph. We leave the analysis of the buildGraph function
as an exercise for you.
+
+
+
diff --git a/pretext/Graphs/BuildingtheKnightsTourGraph.ptx b/pretext/Graphs/BuildingtheKnightsTourGraph.ptx
index f95c9d7c..a329b0cb 100755
--- a/pretext/Graphs/BuildingtheKnightsTourGraph.ptx
+++ b/pretext/Graphs/BuildingtheKnightsTourGraph.ptx
@@ -369,4 +369,7 @@ int main() {
}
+
+
+
diff --git a/pretext/Graphs/BuildingtheWordLadderGraph.ptx b/pretext/Graphs/BuildingtheWordLadderGraph.ptx
index c241d985..1dc89909 100644
--- a/pretext/Graphs/BuildingtheWordLadderGraph.ptx
+++ b/pretext/Graphs/BuildingtheWordLadderGraph.ptx
@@ -248,4 +248,7 @@ int main() {
constructed by the buildGraph function has exactly 53,286 edges, so
the matrix would have only 0.20% of the cells filled! That is a very
sparse matrix indeed.
+
+
+
diff --git a/pretext/Graphs/DepthFirstSearchAnalysis.ptx b/pretext/Graphs/DepthFirstSearchAnalysis.ptx
index 9a1f4c3a..5db6548b 100755
--- a/pretext/Graphs/DepthFirstSearchAnalysis.ptx
+++ b/pretext/Graphs/DepthFirstSearchAnalysis.ptx
@@ -8,5 +8,8 @@
recursively if the vertex is white, the loop will execute a maximum of
once for every edge in the graph or O(E). So, the total time
for depth first search is O(V + E).
+
+
+
diff --git a/pretext/Graphs/DijkstrasAlgorithm.ptx b/pretext/Graphs/DijkstrasAlgorithm.ptx
index d1221d41..b16c46e4 100755
--- a/pretext/Graphs/DijkstrasAlgorithm.ptx
+++ b/pretext/Graphs/DijkstrasAlgorithm.ptx
@@ -239,4 +239,7 @@ vector dijkstra(const vector>& graph, int start) {
+
+
+
diff --git a/pretext/Graphs/DiscussionQuestions.ptx b/pretext/Graphs/DiscussionQuestions.ptx
index 9348199a..a755a5ef 100755
--- a/pretext/Graphs/DiscussionQuestions.ptx
+++ b/pretext/Graphs/DiscussionQuestions.ptx
@@ -171,5 +171,8 @@
algorithm?
+
+
+
diff --git a/pretext/Graphs/GeneralDepthFirstSearch.ptx b/pretext/Graphs/GeneralDepthFirstSearch.ptx
index 9800e489..37f36b50 100644
--- a/pretext/Graphs/GeneralDepthFirstSearch.ptx
+++ b/pretext/Graphs/GeneralDepthFirstSearch.ptx
@@ -344,5 +344,8 @@ main()
DFS Visualization
+
+
+
diff --git a/pretext/Graphs/Glossary.ptx b/pretext/Graphs/Glossary.ptx
index e624e78d..5c111755 100755
--- a/pretext/Graphs/Glossary.ptx
+++ b/pretext/Graphs/Glossary.ptx
@@ -142,5 +142,8 @@
+
+
+
diff --git a/pretext/Graphs/Implementation.ptx b/pretext/Graphs/Implementation.ptx
index 43ac5e1e..be189655 100644
--- a/pretext/Graphs/Implementation.ptx
+++ b/pretext/Graphs/Implementation.ptx
@@ -427,4 +427,7 @@ main()
+
+
+
diff --git a/pretext/Graphs/ImplementingBreadthFirstSearch.ptx b/pretext/Graphs/ImplementingBreadthFirstSearch.ptx
index db1be0b1..ac2926d6 100755
--- a/pretext/Graphs/ImplementingBreadthFirstSearch.ptx
+++ b/pretext/Graphs/ImplementingBreadthFirstSearch.ptx
@@ -434,4 +434,7 @@ int main() {
}
+
+
+
diff --git a/pretext/Graphs/ImplementingKnightsTour.ptx b/pretext/Graphs/ImplementingKnightsTour.ptx
index ada21c57..f336162f 100644
--- a/pretext/Graphs/ImplementingKnightsTour.ptx
+++ b/pretext/Graphs/ImplementingKnightsTour.ptx
@@ -244,7 +244,9 @@ vector runKnightTour(int start, Graph& ktGraph, int n){
return done
-
-
-
+
+
+
+
+
diff --git a/pretext/Graphs/KnightsTourAnalysis.ptx b/pretext/Graphs/KnightsTourAnalysis.ptx
index 86a0ff39..a61911e6 100755
--- a/pretext/Graphs/KnightsTourAnalysis.ptx
+++ b/pretext/Graphs/KnightsTourAnalysis.ptx
@@ -184,4 +184,7 @@ def orderByAvail(n):
+
+
+
diff --git a/pretext/Graphs/Matching.ptx b/pretext/Graphs/Matching.ptx
index 5d55643f..f32cb01c 100644
--- a/pretext/Graphs/Matching.ptx
+++ b/pretext/Graphs/Matching.ptx
@@ -63,4 +63,7 @@
+
+
+
diff --git a/pretext/Graphs/Objectives.ptx b/pretext/Graphs/Objectives.ptx
index cf66a4fe..f9823ec3 100755
--- a/pretext/Graphs/Objectives.ptx
+++ b/pretext/Graphs/Objectives.ptx
@@ -40,4 +40,7 @@
Flowchart outlining the prerequisites for a Computer Science major. The chart begins with 'Math-151' leading into 'CS-220'. Below this, 'CS-150' points to 'CS-151', which in turn branches into 'CS-250' and 'CS-230'. 'CS-230' continues to 'CS-466', and 'CS-151' also points to 'CS-360', which then leads to 'CS-490'. Additionally, 'CS-360' branches off to 'CS-477'. Each course is represented as a node, and the arrows indicate the prerequisite relationship, where one course must be completed before moving on to the next.
+
+
+
diff --git a/pretext/Graphs/PrimsSpanningTreeAlgorithm.ptx b/pretext/Graphs/PrimsSpanningTreeAlgorithm.ptx
index 2dd803ce..220f7563 100755
--- a/pretext/Graphs/PrimsSpanningTreeAlgorithm.ptx
+++ b/pretext/Graphs/PrimsSpanningTreeAlgorithm.ptx
@@ -320,5 +320,8 @@ int prims(const unordered_map>& graph, int start){
+
+
+
diff --git a/pretext/Graphs/ProgrammingExercises.ptx b/pretext/Graphs/ProgrammingExercises.ptx
index 75b6e808..e018876e 100755
--- a/pretext/Graphs/ProgrammingExercises.ptx
+++ b/pretext/Graphs/ProgrammingExercises.ptx
@@ -41,5 +41,8 @@
river.
+
+
+
diff --git a/pretext/Graphs/ShortestPathProblems.ptx b/pretext/Graphs/ShortestPathProblems.ptx
index ec3efd94..a8307c61 100755
--- a/pretext/Graphs/ShortestPathProblems.ptx
+++ b/pretext/Graphs/ShortestPathProblems.ptx
@@ -68,5 +68,8 @@ Routers from One Host to the Next over the Internet
first search, except that here we are concerned with the total weight of
the path rather than the number of hops in the path. It should be noted
that if all the weights are equal, the problem is the same.
+
+
+
diff --git a/pretext/Graphs/StronglyConnectedComponents.ptx b/pretext/Graphs/StronglyConnectedComponents.ptx
index e10c87c7..c2b289e1 100755
--- a/pretext/Graphs/StronglyConnectedComponents.ptx
+++ b/pretext/Graphs/StronglyConnectedComponents.ptx
@@ -131,5 +131,8 @@
The diagram shows a directed graph illustrating the concept of strongly connected components within a network of nodes. Each node is labeled with a letter from A to I and a pair of numbers, which could represent the sequence in which a depth-first search algorithm processed them. The graph is organized with directed edges forming paths between the nodes, suggesting the presence of subgraphs where each node is reachable from every other node within the same subgraph. This kind of representation is commonly used in computer science to illustrate algorithms that identify strongly connected components within a graph, which are maximal sets of vertices with a path to every other vertex in the set.
+
+
+
diff --git a/pretext/Graphs/Summary.ptx b/pretext/Graphs/Summary.ptx
index 407b6f7e..056774d8 100755
--- a/pretext/Graphs/Summary.ptx
+++ b/pretext/Graphs/Summary.ptx
@@ -25,5 +25,8 @@
Minimum weight spanning trees for broadcasting messages.
+
+
+
diff --git a/pretext/Graphs/TheGraphAbstractDataType.ptx b/pretext/Graphs/TheGraphAbstractDataType.ptx
index d0fb6845..e400aee5 100755
--- a/pretext/Graphs/TheGraphAbstractDataType.ptx
+++ b/pretext/Graphs/TheGraphAbstractDataType.ptx
@@ -83,5 +83,8 @@ Reading Question
+
+
+
diff --git a/pretext/Graphs/TheKnightsTourProblem.ptx b/pretext/Graphs/TheKnightsTourProblem.ptx
index a2425120..d8998fe5 100755
--- a/pretext/Graphs/TheKnightsTourProblem.ptx
+++ b/pretext/Graphs/TheKnightsTourProblem.ptx
@@ -26,5 +26,8 @@
visited exactly once.
+
+
+
diff --git a/pretext/Graphs/TheWordLadderProblem.ptx b/pretext/Graphs/TheWordLadderProblem.ptx
index 62bd911c..e20ad0e9 100755
--- a/pretext/Graphs/TheWordLadderProblem.ptx
+++ b/pretext/Graphs/TheWordLadderProblem.ptx
@@ -32,5 +32,8 @@ SAGE
efficient path from the starting word to the ending word.
+
+
+
diff --git a/pretext/Graphs/TopologicalSorting.ptx b/pretext/Graphs/TopologicalSorting.ptx
index 2c7bdb9c..238c432d 100755
--- a/pretext/Graphs/TopologicalSorting.ptx
+++ b/pretext/Graphs/TopologicalSorting.ptx
@@ -67,6 +67,8 @@
Alt text for Figure 29: This image shows the result of a topological sort on a directed acyclic graph representing the steps for making pancakes. The sequence begins with "1 Tbl Oil" at step 17/18, followed by "1 egg" at step 15/16, "3/4 cup milk" at step 1/12, and "1 cup mix" at step 2/11. The next steps are "heat griddle" at step 13/14, "heat syrup" at step 9/10, "pour 1/4 cup" at step 3/8, and "turn when bubbly" at step 4/7. The final action is "eat" at step 5/6. Each step is depicted as an oval, connected by directed paths that indicate the order of operations, with numbers denoting the order in the topological sort.
-
+
+
+
diff --git a/pretext/Graphs/VocabularyandDefinitions.ptx b/pretext/Graphs/VocabularyandDefinitions.ptx
index 134eed8e..8061ced8 100755
--- a/pretext/Graphs/VocabularyandDefinitions.ptx
+++ b/pretext/Graphs/VocabularyandDefinitions.ptx
@@ -95,4 +95,7 @@
-
+
+
+
+
\ No newline at end of file