Transcript Document

Advanced Operations on Graphs
Lecture 6
Cone and Suspension
• The join of G and K1 we call the cone over
G and is denoted by Cone(G) = G*K1.
• The join G*(2K1 ) is called suspension.
Examples
• Any complete multipartite graph is a join of empty
graphs.
• The cone Cone(Cn) is called a pyramid or wheel
Wn.
• Octahedral graph is the suspenstion over C4. It
can be written in the form:
– O3 = (2K1)*(2K1)*(2K1).
• Construction can be generalized to:
– On = (2K1)*(2K1)* ...*(2K1)
Exercises
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N1: Prove that for any graph X at least one of the
graphs X and Xc is connected.
N2: Decribe two graphs G and H, so that H is
isomorphic to an induced subgraph, and also to a
non-induced subgraph of G.
N3: Graph of our original example is isomorphic
to K4-e.
1.
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4.
How many subgraphs has K4-e?
How many non-isomorphic subgraphs has K4-e?
How many induced subgraphs has K4-e?
How many non-isomorphic induced subgraphs has K4-e?
Exercises
• N4. Determine the number of vertices and egdes
of the generalized octahedral graph On.
• N5. Let V = {-1,1}n. Define a graph Gn, whose
vertex set is V and two vertices are adjacent if and
only if d(u,v)2 < n. Prove that Gn is isomorphic to
On.
• N6. Explore the relationship between graphs G1 =
G * (2K1) and G2 = ((G*K1)*K1).
• N7. True or False? The Cone(H) is convex in
Cone(G) if H is convex in G.
• N8. Show that G\H is a spanning subgraph of G.
Homework
• H1. Prove that C8 is isomorphic to its bipartite
complement C8b.
• H2. Determine all paths Pn that are isomorphic to their
bipartite complements Pnb.
• H3. Draw suspension over C5.
• H4. Graph, that is isomorphic to its complement is called
self-complementary. Prove that there exists no selfcomplementary graph on (n+2) vertices, if there exist selfcomplementary graphs on n and on (n+1)-vertices.
• H5. Draw all self-complementary paths and all selfcomplementary cycles.
Subdivision
• Let e 2 E(G) be an edge of G. Let S(G,e) denote the graph
obtained from G by replacing the edge e by a path of
length 2 passing through a new vertex.. Such an operation
is called subdivision of the edge e..
• Let F be a subset of E(G), then S(G,F) denotes the graph
obtained from the subdivision of each edge of F. In the
case F = E, we drop the second argument and S(G)
denotes the subdivision graph G.
• Graph H is a general subdivision of graph G, if H is
obtained from G by a sequence of edge subdivisions..
Graph Homeomorphism
• Graphs G and H are homeomorphic, if they have a
common subdivision.
• Graph G is topologically contained in a graph K, if there
exists a subgraph H of K, that is homeomorphic to G.
Exercises
• N1: Prove that S(G) is bipartite for any simple G.
• N2: Find a general subdivision of K3,3 in G(5,2).
• N3: Given a graph X with n vertices and m edges.
Determine the number of verticexs and the
number of edges of S(X)?
• N4: Graph G is topologically almost contained
in graph K, if there exists a subgraph H of K, that
is a general subdivision of G. Prove that
topological containment implies topological
almost containment and find a counterexample for
the converse.
Matching
• Edges with no common endvertex are called
independent. A set of pairwise independent
edges is called a matching.
Maximal Matching
• A matching that cannot be augmented by
adding new edges is called a maximal
matching.
Perfect Matching
• Proposition: Let M be a matching of a
graph G on n vertices. Then |M| · n/2.
• A matching M with |M| = n/2 is called a
perfect matching.
Exercises
• N5. Prove that |M| · n/2 for any matching.
• N6. Prove that if a graph has a perfect matching
then it has an even number of vertices.
• N7. Prove that a parfect matching is maximal
matching. Give an example of a maximal
matching that is not perfect.
• N8. Show that the family of matchings define an
abstract simplicial complex on E(G).
Abstract Simplicial Complex
a
d
b
c
e
h
f
g
• K µ P(S) is an
abstract simplicial
complex if for each s
2 K and each t µ s it
follows that t 2 K.
• On the left:
• K = {;, a, b, c, d, e, f,
g, h, ab, ad, abd, bc,
be, bcd, bd, ce, df, dg,
de, eh}
Exercises
• N9: Find a perfect matchning in Petersen
graph G(5,2).
• N10: Find a cubic graph with no perfect
matching.
Line Graph L(G)
• Two edges with a common end-vertex are
incident. Incidence is a binary relation on
the edge set E(G).
• Line graph L(G) has the vertex set E(G),
while the edges of L(G) are determined by
the incidence of edges in G.
Examples
• Top row depicts the
Heawood graph and
its fourvalent
linegraph.
• Bottom row depicts
the Petersen graph and
its line graph.
Exercises
• N11: Show that the line graph of a regular graph is regular.
• N12: What is then number of vertices and the number of
edges of L(G), given the number of vertices n and the
number of edges m in G?
• N13: Define trunacated graph T(G) := L(S(G)). Determine
the number of vertices and edges in T(G)?
• N14: Draw T(K4) and T(Q3).
• N15: Define operations L(G) and T(G) only partially, for
some set of vertices or edges, similarly as S(G,F) was
defined. For instance, for bipartite graphs we may define
operations with respect to only one bipartition set.
Variations of Graphs
• Our main topic are simple finite graphs. We
have to be aware of other similar creatures.
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Infinite graphs
Digraphs
General graphs (multigraphs)
Pregraphs
Rooted graphs
...
Infinite Graphs
• An infinite graph may have an infinite set of
vertices or edges. Countable graphs have both sets
coutable (or finite)
• Among countable graphs most tractable are
locally finite graphs. (Each vertex has finite
valence).
• Even more restricted are bouded valence graphs,
where D(G) is finite, in particular regular ones.
Examples
• Examples of infinite graphs:
• Each metric space (X,d) determines a unit distance graph:
» V(X) := X
» x ~ y , d(x,y) = 1.
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Ray or infinite path P1
Double ray or infinite cycle C1
Infinite k-way tree T(1,k).
Infinite square lattice Q1.
Infinite triangular lattice T1.
Infinite hexagonal lattice H1.
Digraphs
• In case of directed graphs (digraphs) there
are no undirected edges e = uv, but there are
directed edges or arcs: a = (u,v).
u
u
e = uv
a = (u,v)
v
v
Digraphs from Binary Relations
• Let R µ V £ V be a binary realtion on V. R
defines in a natural way a digraph.
Exercises
• N16: Following the definion of a category, define a
digraph as a quadruple (V,A,i,t), where i and t are
mappings, that assign to each arc a its intial vertex i(a) and
terminal vertex t(a). Explain the procedure that starts with
a binary relation R and obtains a digraph (V,A,i,t)?
• N17: Define homomorphisms and isomorphisms of
digraphs.
• N18: Formally define a directed path Pn!, directed cycle
Cn! and directed complete graph Kn!.
• N19: Describe a digraph that does not arise from a binary
relation..
• N20: Decribe a procedure that assings to each simple
undirected graph (V,~) a directed graph in such a way that
each undirected path is assigned a pair of oppositely
directed arcs.
Loops, parallel edges,
multigraphs
• A multigraph (general graph) is more
general than a simple graph, since it may
have parallel edges (between vertices u and
v we may have two edges e = uv and f = uv)
or loops. A loop z = uu is na edge that has
both endpoints the same.
u
v
C2, parallel edges
u
C1, loop
Dipoles, bouquets of circles and
more
Q3
B4
The handcuff graph G(1,1)
• Among general graphs
there are several
interesting families.
• A dipole qn has two
vertices and n parallel
edges between them.
• Bouqet of circles Bn has
one vertex with n loops.
• The handcuff graph
G(1,1) is a graph with two
vertices, each one having
an edge and another edge
joining them.
Pregraphs
• If we want to distinguish a loop from a halfedge we need a notion of a pregraph.
Definition of pregraph
• Pregraph X = (V,S,i,r) consists of
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vertex set V,
set of arcs S,
mapping intial vertex, i: S ! V
involution reverse r: S ! S.
• Since r is an involution, we have r2 = 1. The orbits
of r of length 2 are edges, while orbits of length 1
are halfedges.
• A pregraph without half-edges is a general graph.
Rooted Graphs
• Let X be a graph and let r 2 V(X) be a selected
vertex. A pair (X, r) is called a rooted graph
and x is called the root.
• The idea of root can be generalized to a set of
vertices or even a subgraph Y µ X. In some
computer systems the root graph Y corresponds to
the selected part of graph..
• The idea can be generalized to “graduated”
graphs:
• Y0 = r µ Y1 µ ... µ Yk = X.
Edge Contraction
• Let e = uv be an edge of G. By G/e we
denote the graph obtained from G, by
contraction of e. This means we identify
vertices u and v, remove the loop and
possible extra parallel edges.
Exercise
• N21: Prove that K5 can be obtained from
G(5,2) by a series of edge contractions.
Graph Products
• For two graphs G and H we may define
several graphs on the vertex set V(G) £
V(H) that behawe like a product. The most
natural one is the so-called Cartesian
product that we introduce first.
Square Grid Gr(n,m).
• Let Gr(n,m) denote the graph defined by a square
grid in the plane, determined by n  m nodes. We
will call it a square grid. For instance, Gr(2,2) is
isomorphic to C4. We may regard Gr(n,m) as a
product of paths Pn and Pm. In a similar way we
may regards the n-prism graph as the product of a
cycle Cn and K2.
• This is a motivation for the introduction of the
Cartesian product of graphs.
Cartesian Product of Graphs
• Cartesian product G  H of graphs G and H has
the vertex set VG  VH. Two vertices (u,v) and
(u’,v’) are adjacent if and only if:
• u = u’ and v ~ v’ or
• u ~ u’ and v = v’.
More products.
• Cartesian product is not the only product for
graphs. If we define a category of graphs and
maps that are dimension preserving the categorical
product is the so-called tensor product. If,
however, the category admits also graph mappings
that may map edges to vertices, the catregorical
prodiuct is the so-called strong product. We will
meet both of them.
Tensor Product of Graphs
• Tensor product G  H of graphs G and H has the
vertex set VG  VH. Two vertices (u,v) and (u’,v’)
are adjacent if and only if:
• u ~ u’ and v ~ v’.
Strong Product of Graphs
• Strong product G ⊠ H of graphs G and H has the
vertex set VG  VH. Two vertices (u,v) and (u’,v’)
are adjacent if and only if:
– u ~ u’ and v ~ v’ or
– u = u’ and v ~ v’ or
– u ~ u’ and v = v’.
Examples of Products
P5 ¤ P4
P5 ⊠ P4
P5 £ P4
• On the left, there are
cartesian, tensor and
strong products of P5
by P4. [Cartesian
product is the grid.]
Exercises
• N22: Let the graph Gi have ni vertices and mi edges, for i = 1,2. Determine the
number of vertices and edges in their Cartesian product G1 ¤ G2.
• N23: Let the graph Gi have ni vertices and mi edges, for i = 1,2. Determine the
number of vertices and edges in their tensor product product G1 £ G2.
• N24: Let the graph Gi have ni vertices and mi edges, for i = 1,2. Determine the
number of vertices and edges in their strong product G1 £ G2.
• N25: Under what conditions will the Cartesian product G1 ¤ G2 be connected?
• N26: Under what conditions will the tensor product G1 £ G2 be connected?
• N27: Under what conditions will the strong product G1 £ G2 be connected?
• N28: Under what conditions will the Cartesian product G1 ¤ G2 be bipartite?
• N29: Under what conditions will the tensor product G1 £ G2 be bipartite?
• N30: Under what conditions will the strong product G1 £ G2 be bipartite?
Factors
• A spanning subgraph is also called a factor. A kvalent regular factor is simply called a k-factor.
• 1-factor is a different name for a perfect matching.
• A 2-factor is a disjoint union of cycles covering
the vertex set of the graph.
Exercises
• N31. Prove: If a trivalent graph contains a
1-factor, it also has a 2-factor.
• N32. Find a trivalent graph, without a 1factor.
Factorization
• Let G = (V,E) be a graph and let E be disjoint
union of sets E = F1 t F2 t ... t Fs, then Hi = (V,Fi)
are factors of G and the decompostion of G into
these factors is called a factorization. This is
written as:
• G = H1  H2  ...  Hs
If all factors Hi are k-factors, we speak of kfactorization of graph G.
Example.
• For any graph G and any of its spanning
subgraphs H we have factorization:
• G = H  G\H.
Exercises
• N33: Prove that only regular d-valent graphs with d
divisible by k admit a k-factorization.
• N34: Prove that non-regular graphs have no 1factorizations.
• N35: Show that a 2-valent graph G has 1-factorziation, if
and only if it is a disjoint union of even cycles.
• N36: Prove that each prism Pn has a 1-factorization.
• N37. Prove that Petersen graph has no 1-factorization.
• N38. Show that each cubic hamiltonian graph has a 1factorization.
Graph Power
• For a connected graph G and an integer k,
we define the k-th graph power G(k) as
follows:
• V(G(k)) := V(G).
• u ~ v if and only if d(u,v) · k.
Pure Graph Power
• For a connected graph G and an integer k,
we define the k-th pure graph power G[k] as
follows:
• V(G[k]) := V(G).
• u ~ v if and only if d(u,v) = k.
Exercises
• N39. Prove that for any integer k and graph
connected G on n vertices we have:
• G(k) := G[0] = En.
• G(k) = G[0] © G[1] © ... © G[k].
Intermezzo - Partitions, Set
Partitions, Graudated sets, etc.
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Equivalence relation – Set Paritition
Distance Partition – Ordered Set Parition
Graduated Graph – Graduated Set
Valence Sequence – (Number) Partition
Hierarchy – Nested (Graduated) Set Partitions
Rooted Tree, Rooted Graph...
MINIVEGA should support all these structures.
Homework
• H1. Prove that each prism graph
Pn = K2 ¤ Cn admits a 1-factorization.
• H2. Prove that each of the three products
(cartesian, tensor, strong) is associative and
commutative (up to isomoprhism).
• H3. Give a definition of the following
infinite graphs (a suitable drawing suffices):
P1, C1, T(1,k), Q1,T1, H1,