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Induced Matchings in Intersection Graphs
Kathie Camerona
aDepartment of Mathematics
Wilfrid Laurier University
Waterloo, ON
Canada, N2L 3C5
Abstract
An induced matching in a graph G is a set of edges, no two of which meet a com-
mon node or are joined by an edge of G; that is, an induced matching is a matching
which forms an induced subgraph. Induced matchings in graph G correspond pre-
cisely to independent sets of nodes in the square of the line-graph of G; which we
denote by [L(G)]2. Often, if G has a nice representation as an intersection graph,
we can obtain a nice representation of [L(G)]2 as an intersection graph. Then, if
the independent set problem is polytime-solvable in [L(G)]2; the induced matching
problem is polytime-solvable in G:
Key words: induced matching, polygon-circle graph, asteroidal triple-free graph,
interval graph, chordal graph, circular-arc graph, circle graph, bipartite graph,
planar graph, outerplanar graph.
A matching is a set of edges, no two of which are incident. An induced matching
M in a graph G is a matching such that no two edges of M are joined by an
edge of G; that is, an induced matching is a matching which forms an induced
subgraph.
In this paper, we consider the problem of �nding a largest induced matching
in a graph G: This problem is NP-complete for bipartite graphs [2] and forplanar graphs [9].
The line-graph L(G) of graph G has node-set E(G), and an edge joining two
nodes exactly when the edges of G they correspond to are incident. The square
G2 of graph G has node-set V (G), and two nodes are joined inG2 exactly whenthey are joined by an edge or a path of two edges in G:
Remark 1 For any graph G; every induced matching in G is an independent
set of nodes in [L(G)]2; and conversely.
Preprint submitted to Elsevier Preprint 21 May 2000
Thus, to �nd a largest induced matching in a graph G; we can �nd a largest
independent set of nodes in [L(G)]2:
(Note that although line-graphs are claw-free, and there is a polytime algo-
rithm for �nding largest independent sets of nodes in claw-free graphs, squares
of line-graphs need not be claw-free.)
Given a family F of non-empty sets, the intersection graph G(F) of F has
node-set F and an edge between u and v exactly when u \ v 6= ;: There
are many well-studied classes of intersection graphs including the following.
Interval graphs are the intersection graphs of a set of intervals on a line; chordal
graphs are the intersection graphs of a set of subtrees of a tree; circular-arc
graphs are the intersection graph of a set of arcs of a circle; circle graphs are
the intersection graphs of a set of chords of a circle; polygon-circle graphs are
the intersection graphs of a set of (convex) polygons inscribed on a circle.
Polygon-circle graphs include chordal graphs [7], circular-arc graphs [7], circle
graphs, and outerplanar graphs [12].
In [2], it is proved that if G is chordal, [L(G)]2 is chordal. The idea there is that
ifG is the intersection graph of a familyF of subtrees Tv of tree T; then [L(G)]2
is the intersection graph of F 0 = fTu [ Tv : uv 2 E(G) = V ( [L(G)]2)g; which
is a family of subtrees of a tree, since the union of two intersecting subtrees
of a tree is again a subtree. The same idea is easily seen to show that where
G is an interval graph, so is [L(G)]2; and where G is a circular-arc graph, so
is [L(G)]2: There are polytime algorithms for largest independent set of nodes
in chordal graphs [5] and in circular-arc graphs [6], and thus these provide
polytime algorithms for the induced matching problem in these classes.
In this paper, we prove that if G is a polygon-circle graph, then so is [L(G)]2:
This is more diÆcult than for interval, chordal, and circular-arc graphs, be-
cause the union of two intersecting polygons inscribed on a circle is not gen-
erally another polygon inscribed on a circle. The technique used provides a
general approach to determining [L(G)]2 for an intersection graph G. F. Gavril
[4] has given a polytime algorithm for �nding a largest independent set of nodes
in a polygon-circle graph, so this provides a polytime algorithm for �nding a
largest induced matching in polygon-circle graphs.
Note that not all subclasses of polygon-circle graphs have the property that
for G in the subclass, [L(G)]2 is also in the subclass - this is not true for the
outerplanar graph C5 for instance.
An independent set of three vertices is called an asteroidal triple if between
each pair in the triple there exists a path that avoids the neighborhood of the
third. A graph is asteroidal triple-free (AT-free) if it contains no asteroidal
triple. AT-free graphs are not yet known to be the intersection graphs of some
nice family, however they contain several classes of intersection graphs includ-
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ing interval graphs [8] and cocomparability graphs [3]. In [11], it is proved
that if G is AT-free, then so is L(G): In [13], it is proved that if H is AT-free,
then so is H2: It follows that if G is AT-free, so is [L(G)]2: In [1] and [10],
polytime algorithms are given for �nding a largest independent set of nodes in
AT-free graphs. Thus these algorithms provide polytime algorithms for �nding
a largest induced matching in AT-free graphs.
References
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[10] Ekkehard G�unther K�ohler, Graphs Without Asteroidal Triples, PhD thesis,Technical University of Berlin, Cuvillier Verlag, G�ottingen,199_9
[11] Ekkehard G�unther K�ohler and Matthias Kriesell, Edge-Dominating trails inAT-free graphs,Technical University of Berlin TechReport-615-1998, 1998.
[12] Alexandr Kostochka and Jan Kratochv��l, Covering and coloring polygon-circlegraphs, Discrete Mathematics 163 (1997), 299{305.
[13] A. Raychaudhuri, On powers of interval and unit interval graphs, CongressusNumerantium 59 (1987), 235-242.
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