indifference graph

{{Short description|Intersection graph of unit intervals on the real number line}}

File:Indifference graph.svg

In graph theory, a branch of mathematics, an indifference graph is an undirected graph constructed by assigning a real number to each vertex and connecting two vertices by an edge when their numbers are within one unit of each other.{{citation

| last = Roberts | first = Fred S. | author-link = Fred S. Roberts

| contribution = Indifference graphs

| mr = 0252267

| pages = 139–146

| publisher = Academic Press, New York

| title = Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968)

| year = 1969}}. Indifference graphs are also the intersection graphs of sets of unit intervals, or of properly nested intervals (intervals none of which contains any other one). Based on these two types of interval representations, these graphs are also called unit interval graphs or proper interval graphs; they form a subclass of the interval graphs.

Equivalent characterizations

File:Forbidden indifference subgraphs.svg for the indifference graphs: the claw, sun, and net (top, left–right) and cycles of length four or more (bottom)]]

The finite indifference graphs may be equivalently characterized as

| last1 = Bogart | first1 = Kenneth P.

| last2 = West | first2 = Douglas B. | author2-link = Douglas West (mathematician)

| doi = 10.1016/S0012-365X(98)00310-0

| issue = 1-3

| journal = Discrete Mathematics

| mr = 1687858

| pages = 21–23

| title = A short proof that "proper = unit"

| volume = 201

| year = 1999| arxiv = math/9811036

}}.

  • The claw-free interval graphs,
  • The graphs that do not have an induced subgraph isomorphic to a claw K1,3, net (a triangle with a degree-one vertex adjacent to each of the triangle vertices), sun (a triangle surrounded by three other triangles that each share one edge with the central triangle), or hole (cycle of length four or more),{{citation

| last = Wegner | first = G.

| location = Göttingen, Germany

| publisher = Göttingen University

| series = Ph.D. thesis

| title = Eigenschaften der Nerven homologisch-einfacher Familien im Rn

| year = 1967}}. As cited by {{harvtxt|Hell|Huang|2004}}.

| last1 = Looges | first1 = Peter J.

| last2 = Olariu | first2 = Stephan

| doi = 10.1016/0898-1221(93)90308-I

| issue = 7

| journal = Computers & Mathematics with Applications

| mr = 1203643

| pages = 15–25

| title = Optimal greedy algorithms for indifference graphs

| volume = 25

| year = 1993| doi-access = free

}}.

  • The graphs with no astral triple, three vertices connected pairwise by paths that avoid the third vertex and also do not contain two consecutive neighbors of the third vertex,{{citation

| last = Jackowski | first = Zygmunt

| doi = 10.1016/0012-365X(92)90135-3

| issue = 1-3

| journal = Discrete Mathematics

| mr = 1180196

| pages = 103–109

| title = A new characterization of proper interval graphs

| volume = 105

| year = 1992| doi-access = free

}}.

  • The graphs in which each connected component contains a path in which each maximal clique of the component forms a contiguous sub-path,{{citation

| last1 = Gutierrez | first1 = M.

| last2 = Oubiña | first2 = L.

| doi = 10.1002/(SICI)1097-0118(199602)21:2<199::AID-JGT9>3.0.CO;2-M

| issue = 2

| journal = Journal of Graph Theory

| mr = 1368745

| pages = 199–205

| title = Metric characterizations of proper interval graphs and tree-clique graphs

| volume = 21

| year = 1996}}.

  • The graphs whose vertices can be numbered in such a way that every shortest path forms a monotonic sequence,
  • The graphs whose adjacency matrices can be ordered such that, in each row and each column, the nonzeros of the matrix form a contiguous interval adjacent to the main diagonal of the matrix.{{citation

| last = Mertzios | first = George B.

| doi = 10.1016/j.aml.2007.04.001

| issue = 4

| journal = Applied Mathematics Letters

| mr = 2406509

| pages = 332–337

| title = A matrix characterization of interval and proper interval graphs

| volume = 21

| year = 2008

}}.

  • The induced subgraphs of powers of chordless paths.{{citation

| last1 = Brandstädt | first1 = Andreas

| last2 = Hundt | first2 = Christian

| last3 = Mancini | first3 = Federico

| last4 = Wagner | first4 = Peter

| doi = 10.1016/j.disc.2009.10.006

| journal = Discrete Mathematics

| pages = 897–910

| title = Rooted directed path graphs are leaf powers

| volume = 310

| year = 2010| doi-access = free

}}.

  • The leaf powers having a leaf root which is a caterpillar.

For infinite graphs, some of these definitions may differ.

Properties

Because they are special cases of interval graphs, indifference graphs have all the properties of interval graphs; in particular they are a special case of the chordal graphs and of the perfect graphs. They are also a special case of the circle graphs, something that is not true of interval graphs more generally.

In the Erdős–Rényi model of random graphs, an n-vertex graph whose number of edges is significantly less than n^{2/3} will be an indifference graph with high probability, whereas a graph whose number of edges is significantly more than n^{2/3} will not be an indifference graph with high probability.{{citation

| last = Cohen | first = Joel E.

| doi = 10.1016/0012-365X(82)90184-4

| issue = 1

| journal = Discrete Mathematics

| mr = 676708

| pages = 21–24

| title = The asymptotic probability that a random graph is a unit interval graph, indifference graph, or proper interval graph

| volume = 40

| year = 1982| doi-access = free

}}.

The bandwidth of an arbitrary graph G is one less than the size of the maximum clique in an indifference graph that contains G as a subgraph and is chosen to minimize the size of the maximum clique.{{citation

| last1 = Kaplan | first1 = Haim

| last2 = Shamir | first2 = Ron

| doi = 10.1137/S0097539793258143

| issue = 3

| journal = SIAM Journal on Computing

| mr = 1390027

| pages = 540–561

| title = Pathwidth, bandwidth, and completion problems to proper interval graphs with small cliques

| volume = 25

| year = 1996}}. This property parallels similar relations between pathwidth and interval graphs, and between treewidth and chordal graphs. A weaker notion of width, the clique-width, may be arbitrarily large on indifference graphs.{{citation

| last1 = Golumbic | first1 = Martin Charles | author1-link = Martin Charles Golumbic

| last2 = Rotics | first2 = Udi

| contribution = The clique-width of unit interval graphs is unbounded

| mr = 1745205

| pages = 5–17

| series = Congressus Numerantium

| title = Proceedings of the Thirtieth Southeastern International Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1999)

| volume = 140

| year = 1999}}. However, every proper subclass of the indifference graphs that is closed under induced subgraphs has bounded clique-width.{{citation

| last = Lozin | first = Vadim V.

| contribution = From tree-width to clique-width: excluding a unit interval graph

| doi = 10.1007/978-3-540-92182-0_76

| mr = 2539978

| pages = 871–882

| publisher = Springer, Berlin

| series = Lecture Notes in Comput. Sci.

| title = Algorithms and computation

| volume = 5369

| year = 2008}}.

Every connected indifference graph has a Hamiltonian path.{{citation

| last = Bertossi | first = Alan A.

| doi = 10.1016/0020-0190(83)90078-9

| issue = 2

| journal = Information Processing Letters

| mr = 731128

| pages = 97–101

| title = Finding Hamiltonian circuits in proper interval graphs

| volume = 17

| year = 1983}}. An indifference graph has a Hamiltonian cycle if and only if it is biconnected.{{citation

| last1 = Panda | first1 = B. S.

| last2 = Das | first2 = Sajal K.

| doi = 10.1016/S0020-0190(03)00298-9

| issue = 3

| journal = Information Processing Letters

| mr = 1986780

| pages = 153–161

| title = A linear time recognition algorithm for proper interval graphs

| volume = 87

| year = 2003}}.

Indifference graphs obey the reconstruction conjecture: they are uniquely determined by their vertex-deleted subgraphs.{{citation

| last = von Rimscha | first = Michael

| doi = 10.1016/0012-365X(83)90099-7

| issue = 2-3

| journal = Discrete Mathematics

| mr = 724667

| pages = 283–291

| title = Reconstructibility and perfect graphs

| volume = 47

| year = 1983| doi-access = free

}}.

Algorithms

As with higher dimensional unit disk graphs, it is possible to transform a set of points into their indifference graph, or a set of unit intervals into their unit interval graph, in linear time as measured in terms of the size of the output graph. The algorithm rounds the points (or interval centers) down to the nearest smaller integer, uses a hash table to find all pairs of points whose rounded integers are within one of each other (the fixed-radius near neighbors problem), and filters the resulting list of pairs for the ones whose unrounded values are also within one of each other.{{citation

| last1 = Bentley | first1 = Jon L. | author1-link = Jon Bentley (computer scientist)

| last2 = Stanat | first2 = Donald F.

| last3 = Williams | first3 = E. Hollins Jr.

| doi = 10.1016/0020-0190(77)90070-9

| issue = 6

| journal = Information Processing Letters

| mr = 0489084

| pages = 209–212

| title = The complexity of finding fixed-radius near neighbors

| volume = 6

| year = 1977}}.

It is possible to test whether a given graph is an indifference graph in linear time, by using PQ trees to construct an interval representation of the graph and then testing whether a vertex ordering derived from this representation satisfies the properties of an indifference graph. It is also possible to base a recognition algorithm for indifference graphs on chordal graph recognition algorithms. Several alternative linear time recognition algorithms are based on breadth-first search or lexicographic breadth-first search rather than on the relation between indifference graphs and interval graphs.{{citation

| last1 = Corneil | first1 = Derek G. | author1-link = Derek Corneil

| last2 = Kim | first2 = Hiryoung

| last3 = Natarajan | first3 = Sridhar

| last4 = Olariu | first4 = Stephan

| last5 = Sprague | first5 = Alan P.

| doi = 10.1016/0020-0190(95)00046-F

| issue = 2

| journal = Information Processing Letters

| mr = 1344787

| pages = 99–104

| title = Simple linear time recognition of unit interval graphs

| volume = 55

| year = 1995| citeseerx = 10.1.1.39.855}}.{{citation

| last1 = Herrera de Figueiredo | first1 = Celina M.

| last2 = Meidanis | first2 = João

| last3 = Picinin de Mello | first3 = Célia

| doi = 10.1016/0020-0190(95)00133-W

| issue = 3

| journal = Information Processing Letters

| mr = 1365411

| pages = 179–184

| title = A linear-time algorithm for proper interval graph recognition

| volume = 56

| year = 1995}}.{{citation

| last = Corneil | first = Derek G. | author-link = Derek Corneil

| doi = 10.1016/j.dam.2003.07.001

| issue = 3

| journal = Discrete Applied Mathematics

| mr = 2049655

| pages = 371–379

| title = A simple 3-sweep LBFS algorithm for the recognition of unit interval graphs

| volume = 138

| year = 2004| doi-access = free

}}.{{citation

| last1 = Hell | first1 = Pavol | author1-link = Pavol Hell

| last2 = Huang | first2 = Jing

| year = 2004

| doi = 10.1137/S0895480103430259

| issue = 3

| journal = SIAM Journal on Discrete Mathematics

| mr = 2134416

| pages = 554–570

| title = Certifying LexBFS recognition algorithms for proper interval graphs and proper interval bigraphs

| volume = 18}}.

Once the vertices have been sorted by the numerical values that describe an indifference graph (or by the sequence of unit intervals in an interval representation) the same ordering can be used to find an optimal graph coloring for these graphs, to solve the shortest path problem, and to construct Hamiltonian paths and maximum matchings, all in linear time. A Hamiltonian cycle can be found from a proper interval representation of the graph in time O(n\log n), but when the graph itself is given as input, the same problem admits linear-time solution that can be generalized to interval graphs.{{citation

| last = Keil | first = J. Mark

| doi = 10.1016/0020-0190(85)90050-X

| issue = 4

| journal = Information Processing Letters

| mr = 801816

| pages = 201–206

| title = Finding Hamiltonian circuits in interval graphs

| volume = 20

| year = 1985}}.{{citation

| last = Ibarra | first = Louis

| doi = 10.1016/j.ipl.2009.07.010

| issue = 18

| journal = Information Processing Letters

| mr = 2552898

| pages = 1105–1108

| title = A simple algorithm to find Hamiltonian cycles in proper interval graphs

| volume = 109

| year = 2009}}.

List coloring remains NP-complete even when restricted to indifference graphs.{{citation

| last = Marx | first = Dániel

| doi = 10.1016/j.dam.2005.10.008

| issue = 6

| journal = Discrete Applied Mathematics

| mr = 2212549

| pages = 995–1002

| title = Precoloring extension on unit interval graphs

| volume = 154

| year = 2006| doi-access = free

}}. However, it is fixed-parameter tractable when parameterized by the total number of colors in the input.

Applications

In mathematical psychology, indifference graphs arise from utility functions, by scaling the function so that one unit represents a difference in utilities small enough that individuals can be assumed to be indifferent to it.

In this application, pairs of items whose utilities have a large difference may be partially ordered by the relative order of their utilities, giving a semiorder.{{citation

| last = Roberts | first = Fred S. | author-link = Fred S. Roberts

| journal = Journal of Mathematical Psychology

| mr = 0258486

| pages = 243–258

| title = On nontransitive indifference

| volume = 7

| year = 1970

| doi=10.1016/0022-2496(70)90047-7}}.

In bioinformatics, the problem of augmenting a colored graph to a properly colored unit interval graph can be used to model the detection of false negatives in DNA sequence assembly from complete digests.{{citation

| last1 = Goldberg | first1 = Paul W.

| last2 = Golumbic | first2 = Martin C.

| last3 = Kaplan | first3 = Haim

| last4 = Shamir | first4 = Ron

| doi = 10.1089/cmb.1995.2.139

| issue = 2

| journal = Journal of Computational Biology

| pmid = 7497116

| title = Four strikes against physical mapping of DNA

| volume = 2

| year = 2009}}.

See also

  • Threshold graph, a graph whose edges are determined by sums of vertex labels rather than differences of labels
  • Trivially perfect graph, interval graphs for which every pair of intervals is nested or disjoint rather than properly intersecting
  • Unit disk graph, a two-dimensional analogue of the indifference graphs

References

{{reflist|30em}}