8 'סמ לוגרתcs4cyber.wdfiles.com/local--files/recitations/rec8_final.pdf · Graph representation •A graph is a set of vertices or nodes connected by edges •A graph may be

מבוא כללי לתכנות ולמדעי המחשב

8'תרגול מס

דין שמואל[email protected]

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מבוא לתורת הגרפים

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Graph representation

• A graph is a set of vertices or nodes connected by edges

• A graph may be directed or undirected

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• We will represent graphs using a 2-dimentional list:

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(0) (1) (2) (3)

(0)

(1)

(2)

(3)

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• We will represent graphs using a 2-dimentional list:

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(0) (1) (2) (3)

(0) 0 2 2 1

(1) 2 0 0 1

(2) 2 0 0 1

(3) 1 1 1 0

>>> g = [[0,2,2,1], [2,0,0,1], [2,0,0,1], [1,1,1,0]]

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Undirected graph

• Can this matrix represent an undirected graph?

>>> g = [[0,1,0], [1,1,1], [0,0,0]]

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Undirected graph

• Can this matrix represent an undirected graph?

• No! an undirected graph must be represented

by a symmetric matrix:• For every entry: g[i][j] = g[j][i]

>>> g = [[0,1,0], [1,1,1], [0,0,0]]

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exercise

• Assume in a graph g, for every entry: g[i][j] = g[j][i].

Does it necessarily mean that g is undirected?

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exercise

• Write a function called is_symmetric

• Argument:• graph – a 2-dimentional list representing a graph

• Returns True if the graph is (possibly) undirected, i.e.,For every entry: g[i][j] = g[j][i]

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exercise

• Write a function called is_symmetric

• Argument:• graph – a 2-dimentional list representing a graph

• Returns True if the graph is (possibly) undirected, i.e.,For every entry: g[i][j] = g[j][i]

def is_symmetric(g):

n = len(g) #number of nodes in the graph

for i in range(n):

for j in range(n):

if g[i][j] != g[j][i]: #not symmetric!

return False

return True

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Bipartite graph

• Definition: an undirected bipartite graph is a graph with the following properties:

• Its vertices can be divided into two disjoint sets 𝑈, 𝑉

so that every edge connects a vertex in 𝑈 to one in 𝑉

• No edges between two vertices in U

• No edges between two vertices in V

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cliques

• Definition: a clique is a graph where every two vertices are connected by an edge

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cliques

• Definition: a clique is a graph where every two vertices are connected by an edge

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exercise

• Question: a graph is a clique the graph is bipartite?

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exercise


• NO

• Proof: Consider a clique of 3 vertices G

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exercise


• NO


• Divide G into 2 disjoint sets U,V

UV

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exercise


• NO



• Observe: exactly one subset (either U or V)has at least two vertices, call two of these vertices 𝑣1, 𝑣2

UV

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exercise


• NO




• As all vertices are connected, so are 𝑣1, 𝑣2!

UV

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exercise


• NO




• As all vertices are connected, so are 𝑣1, 𝑣2!

• Alternately, G contains an odd length cycle

UV

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exercise

• In a house full of ginger and grey cats• Every ginger cat screams if it sees another ginger cat.

• Every grey cat screams if it sees another grey cat

• Can we spread the cats around the house so that no 2 cats of the same color stay in adjacent rooms, and avoid the screaming?

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exercise

• In a house full of ginger and grey cats• Every ginger cat screams if it sees another ginger cat.

• Every grey cat screams if it sees another grey cat

• Can we spread the cats around the house so that no 2 cats of the same color stay in adjacent rooms, and avoid the screaming?

• We will build a graph where every room is a vertex.

• Every 2 vertices will be connected with an edge if the 2 rooms are adjacent.

• Check if we can divide the graph into 2 disjoint sets of ‘rooms’, for gingers and for greys.

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exercise

• Prove: A graph is bipartite ⇔ its vertices can be colored in two colors such that there is no edge connecting two vertices of the same color

תנאי מספיק והכרחי לכך שניתן לצבוע את קודקודי -היות הגרף דו צדדי: כלומר•מאותו הצבעקודקודיםהגרף בשני צבעים כך ששום קשת לא מחברת

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exercise

• Prove: A graph is bipartite its vertices can be colored in two colors such that there is no edge connecting two vertices of the same color

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exercise


• Proof:

• The graph is bipartite, lets divide its vertices to disjoint

Sets.

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exercise


• Proof:

• The graph is bipartite, lets divide its vertices to disjoint

Sets.

• We will color each set with different color

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exercise


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exercise


• Proof:

• A graph can be colored with red and blue such that

There is no edge connecting blue vertex to a red one

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exercise


• Proof:



• Lets divide the vertices into 2 sets:• R= all the red vertices

• B = all the blue vertices

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exercise


• Proof:



• Lets divide the vertices into 2 sets:• R= all the red vertices• B = all the blue vertices

• no edge connects red vertex to a blue one

• The graph is bipartite with R,B partition.

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אוילרמעגל

:מהשיעור•

.בכל קשת בדיוק פעם אחתבגרף הוא מסלול שעובר אוילרמסלול •

אוילרמעגל זהו , אם מסלול כזה מסתיים באותו צומת בו הוא התחיל•

:אוילרמשפט •

.כל הצמתים בעלי דרגה זוגית: אוילרמעגלתנאי מספיק והכרחי לקיום •

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אוילרמסלול

:המשך-אוילרמשפט •

בעלי 2כל הצמתים למעט בדיוק : אוילרמסלולתנאי מספיק והכרחי לקיום •.דרגה זוגית

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:אינטואיציה•

-אוילרגרף עם מעגל נקח•

.דרגת כל הצמתים היא זוגית•

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.דרגת כל הצמתים היא זוגית•נוכל להתחיל את , אוילרכיוון וזה מעגל •

המסלול בכל קשת שנרצה

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נמחק את הקשת האחרונה במעגל•

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בעלי דרגה 2כל הצמתים למעט בדיוק : אוילרמסלולתנאי מספיק והכרחי לקיום •.זוגית





.נמחק את הקשת האחרונה במעגל•

שחוברו על ידי הקשת הזוהקודקודים•

.כעת בעלי דרגה אי זוגית, (ורק הם)

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אוילרמעגל \מסלול

• Lets write a function euler

• Argument:• A graph

• Prints ‘Euler cycle’ if the graph contains an Euler cycle

• Prints ‘Euler path’ if the graph contains an Euler path

• Prints ‘No Euler path’ otherwise.

• Assume G is undirected, connected and has no self edges

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• Lets write a function euler

• Argument:• A graph

• Prints ‘Euler cycle’ if the graph contains an Euler cycle

• Prints ‘Euler path’ if the graph contains an Euler path

• Prints ‘No Euler path’ otherwise.

• Assume G is undirected, connected and has no self edges

• Solution: count how many vertices have an odd degree.

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def euler (G):

n = len(G)

odd = 0

for i in range(n):

deg = sum(G[i])

if deg%2 == 1:

odd+=1

if odd == 0:

print(“Euler cycle”)

elif odd == 2:

print(“Euler path”)

else:

print(“No Euler path”)38


def euler (G):

n = len(G)

odd = 0

for i in range(n):

deg = sum(G[i])

if deg%2 == 1:

odd+=1

if odd == 0:


elif odd == 2:


else:

print(“No Euler path”)

Number of nodes in G

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def euler (G):

n = len(G)

odd = 0

for i in range(n):

deg = sum(G[i])

if deg%2 == 1:

odd+=1

if odd == 0:


elif odd == 2:


else:

print(“No Euler path”)

Number of nodes in G

Number of edges that ‘touch’ node i

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• Can you draw the ‘House of Santa’ without raising your pencil?

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• Can you draw the ‘House of Santa’ without raising your pencil?

• Or: does the ‘House of Santa’ graph have an Euler path?

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Documents

8 'סמ לוגרתcs4cyber.wdfiles.com/local--files/recitations/rec8_final.pdf · Graph representation •A graph is a set of vertices or nodes connected by edges •A graph may be