Appendix A Graph Problems

In this appendix, we give definitions of the graph problems that are used in this thesis. Most graph problems are defined both for simple graphs and for multigraphs. If the problems are mentioned in Garey and Johnson [1979], we give the number of the problem in this book between square brackets. For almost all of the problems defined in this appendix, there are two decision variants and one optimization variant. In the first decision variant, say PROBLEM, the instance consists of a graph G andanintegerk, and we ask whether a solution exists for the graph which has valueatleastoratmostk. In the second decision variant, the integer is taken to be some constant k, and the instance consists of a graph only. The question is the same as for the first decision variant, and the problem is denoted by k-PROBLEM for any fixed integer k.Inthe optimization variant, the instance is a graph, and we ask for the maximum or minimum value of the integer k for which the decision problem has a ‘yes’ answer. The problem is denoted by MAX PROBLEM or MIN PROBLEM, respectively (unless stated otherwise). In this appendix, we only give definitions of the first variant of the decision problem: the definitions of the other two variants follow directly. Each of the defined problems has a constructive version, which follows directly from the context. For definitions of tree decompositions and treewidth, see Definition 2.2.1. For path de- compositions and pathwidth, see Definition 2.2.2.

TREEWIDTH

;   Instance: AgraphG =V E ,integerk 1. Question: Does G have treewidth at most k,i.e.doesG have a tree decomposition of width at most k?

PATHWIDTH

;   Instance: AgraphG =V E ,integerk 1. Question: Does G have pathwidth at most k,i.e.doesG have a path decomposition of width at most k? A Hamiltonian circuit in a graph G is a simple cycle in G containing all vertices of G.A Hamiltonian path in a graph G is a path in G containing all vertices of G.

HAMILTONIAN CIRCUIT [GT37]

;  Instance: AgraphG =V E . Question: Does G contain a Hamiltonian circuit?

217 Appendix A Graph Problems

HAMILTONIAN PATH [GT39]

;  Instance: AgraphG =V E .

Question: Does G contain a Hamiltonian path?

  An independent set of a graph G is a set W V G such that no two vertices in W are adjacent in G.

INDEPENDENT SET [GT20]

;   Instance: AgraphG =V E ,integerk 0.

Question: Does G have an independent set of cardinality at least k?

 ;::: ;    For any integer k  1, a k-coloring of a graph G is a partition V1 Vk of V G such that for each i, Vi is an independent set of G.

COLORABILITY [GT4]

;   Instance: AgraphG =V E ,anintegerk 1. Question: Does G have a k-coloring? The minimization problem in which we ask for a minimum value of k for which a k-coloring exists, is denoted by CHROMATIC NUMBER. For definitions of layouts and bandwidth, see Definition 2.3.2.

BANDWIDTH [GT40]

;  Instance: AgraphG =V E ,anintegerk. Question: Does G have bandwidth at most k, i.e. does G have a layout of bandwidth at most k?

The following problem is defined for all integers d  0.

INDUCED d-DEGREE SUBGRAPH

;  

Instance: AgraphG =V E ,integerk 0.

[ ]

Question: Is there a set S V such that all vertices in G S have degree at most p,and j jS k?

For d = 0, this is the INDEPENDENT SET problem.

VERTEX COVER [GT1]

;  

Instance: AgraphG =V E ,anintegerk 1.

f ; g2   2 2

Question: Is there a set S V such that for each edge v w E G , v S or w S,and j jS k?

The following problem is defined for any fixed integer p  1.

p-DOMINATING SET

;  

Instance: AgraphG =V E ,anintegerk 1.

Question: Is there a set S V such that all vertices in V S have at least p neighbors in S, j and jS k?.

For p = 1, this is the DOMINATING SET problem, numbered [GT2].

;   ;  A cut in a graph G =V E is a partition V1 V2 of V.

218

LARGE CUT [ND16]

;  

Instance: AgraphG =V E ,integerk 1.

;  jff ; g2 j 2 ^ 2 gj  Question: Does G have a cut V1 V2 such that v w E v V1 w V2 k? The corresponding maximization problem is called MAX CUT.

PARTITION INTO CLIQUES [GT15]

;  

Instance: AgraphG =V E ,anintegerk 1.

;::: ;    [ ] Question: Is there a partition V1 Vs of V in which for each i,1 i s, G Vi is a

complete graph, and s  k?

COVERING BY CLIQUES [GT17]

;  

Instance: AgraphG =V E ,anintegerk 1.

;::: ; g   [ ]

Question: Is there a set fV1 Vs , in which for each i,1 i s, Vi V , G Vi is a

  2  [ ] complete graph, and for each edge e 2 E,thereisani,1 i s, such that e E G Vi ,and

furthermore, s  k?

HAMILTONIAN CIRCUIT COMPLETION [GT34]

;  

Instance: AgraphG =V E ,anintegerk 0.

0

ff ; gj ; 2 g ; [ 

Question: Is there a set F u v u v V , such that G =V E F contains a Hamil- j tonian circuit, and jF k?

HAMILTONIAN PATH COMPLETION

;  

Instance: AgraphG =V E ,anintegerk 0.

0

= ; gj ; 2 g ; [ 

Question: Is there a set F ffu v u v V , such that G V E F contains a Hamil- j

tonian path, and jF k?.

;  = ;  A spanning tree of a graph G =V E is a subgraph T V F of G which is a tree.

LEAF SPANNING TREE [ND2]

;   Instance: AgraphG =V E ,anintegerk 1. Question: Is there a spanning tree of G in which at least k vertices have degree one?

LONG PATH [ND29]

;   Instance: AgraphG =V E ,anintegerk 1. Question: Does G have a path of length at least k?

The corresponding maximization problem is called LONGEST PATH.

LONG CYCLE [ND28]

;   Instance: AgraphG =V E ,anintegerk 1.

Question: Does G have a cycle of length k  1?

The corresponding maximization problem is called LONGEST CYCLE. The following seven problems are only used in Chapter 4. For definitions of sandwich graphs, k-intervalizations, k-unit-intervalizations, and the (proper) pathwidth and bandwidth of sandwich graphs, see Section 4.1 and Section 4.3.

219 Appendix A Graph Problems

INTERVALIZING SANDWICH GRAPHS (ISG)

; ;   Instance: A sandwich graph S =V E1 E2 ,anintegerk 1. Question: Is there a k-intervalization of S?

INTERVALIZING COLORED GRAPHS (ICG)

;   Instance: A simple graph G =V E ,anintegerk 1andak-coloring c for G. Question: Is there a k-intervalization of G and c?

UNIT-INTERVALIZING SANDWICH GRAPHS (UISG)

; ;   Instance: A sandwich graph S =V E1 E2 ,anintegerk 1. Question: Is there a k-unit-intervalization of S?

UNIT-INTERVALIZING COLORED GRAPHS (UICG)

;   Instance: A simple graph G =V E ,anintegerk 1andak-coloring c for G. Question: Is there a k-unit-intervalization of G and c?

SANDWICH PATHWIDTH

; ;   Instance: A sandwich graph S =V E1 E2 ,anintegerk 1.

Question: Does S have pathwidth at most k 1?

SANDWICH PROPER PATHWIDTH

; ;   Instance: A sandwich graph S =V E1 E2 ,anintegerk 1.

Question: Does S have proper pathwidth at most k 1, i.e. is there a proper path decompo- sition of S?

SANDWICH BANDWIDTH

; ;   Instance: A sandwich graph S =V E1 E2 ,integerk 1.

Question: Does S have bandwidth at most k 1, i.e. is there a legal layout of bandwidth at

most k 1ofS? The following two problems are only used in Chapter 8. For definitions of source-sink labeled graphs, series-parallel graphs and sp-trees, see Section 2.3.3.

SOURCE-SINK LABELED SERIES-PARALLEL GRAPH

; ; 

Instance: A source-sink labeled multigraph G s t .

; ;   ; ;  Question: Is G s t series-parallel, i.e. is there an sp-tree for G s t ? For directed input graphs, this problem is denoted by DLSPG, for indirected input graphs by LSPG. SERIES-PARALLEL GRAPH Instance: A multigraph G. Question: Is G series-parallel, i.e. is there an sp-tree for G? For directed input graphs, this problem is denoted by DSPG, for indirected input graphs by SPG. The following problem is only used in Chapter 9 (for definitions of a labeled multigraph and the treewidth of such a graph, see Section 9.1).

220 TREEWIDTH AT MOST TWO (TW2) Instance: A connected labeled multigraph G. Question: Does G have treewidth at most two, i.e. is there a tree decomposition of width at most two of G?

221 Appendix A Graph Problems

222