On the strong chromatic index and maximum induced matching of tree-cographs and permutation graphs

Equivalently, a strong edge coloring of G is a vertex coloring of L(G) 2 , the square of the linegraph of G. The strong chromatic index of G is the minimal integer k such that G has a strong edge coloring with k colors. We denote the strong chromatic index of G by sχ ′ (G).

The class of tree-cographs was introduced by Tinhofer in [16].

Definition 2. Tree-cographs are defined recursively by the following rules.

1. Every tree is a tree-cograph.

2. If G is a tree-cograph then also the complement Ḡ of G is a tree-cograph.

3. For k 2, if G 1 , . . . , G k are connected tree-cographs then also the disjoint union is a tree-cograph.

Let G be a tree-cograph. A decomposition tree for G consists of a rooted binary tree T in which each internal node, including the root, is labeled as a join node ⊗ or a union node ⊕. The leaves of T are labeled by trees or complements of trees. It is easy to see that a decomposition tree for a tree-cograph G can be obtained in O(n 3 ) time.

⋆ National Science Council of Taiwan Support Grant NSC 99-2218-E-007-016.

The linegraph L(G) of a graph G is the intersection graph of the edges of G [1]. It is well-known that, when G is a tree then the linegraph L(G) of G is a clawfree blockgraph [11]. A graph is chordal if it has no induced cycles of length more than three [7]. Notice that blockgraphs are chordal.

A vertex x in a graph G is simplicial if its neighborhood N(x) induces a clique in G. Chordal graphs are characterized by the property of having a perfect elimination ordering, which is an ordering [v 1 , . . . , v n ] of the vertices of G such that v i is simplicial in the graph induced by {v i , . . . , v n }. A perfect elimination ordering of a chordal graph can be computed in linear time [15]. This implies that chordal graphs have at most n maximal cliques, and the clique number can be computed in linear time.

). If G is a chordal graph then L(G) 2 is also chordal.

Proof. Any chordal graph is the intersection graph of a collection of subtrees of a tree. Let G be the intersection graph of a collection of subtrees of a tree. An intersection model for L(G) 2 is obtained by taking the union of every pair of intersecting subtrees.

⊓ ⊔

). Let k ∈ N and let k 4. Let G be a graph and assume that G has no induced cycles of length at least k. Then L(G) 2 has no induced cycles of length at least k.

Proof. Let G be a tree-cograph.

First observe that trees are bipartite. It follows that complements of trees have no induced cycles of length more than four.

We prove the claim by induction on the depth of a decomposition tree for G. If G is the union of two tree-cographs G 1 and G 2 then the claim follows by induction since any induced cycle is contained in one of G 1 and G 2 . Assume G is the join of two tree-cographs G 1 and G 2 . Assume that G has an induced cycle C of length at least five. We may assume that C has at least one vertex in each of G 1 and G 2 . If one of G 1 and G 2 has more than two vertices of C, then C has a vertex of degree at least three, which is a contradiction.

⊓ ⊔ Lemma 2. Let T be a tree. Then L( T ) 2 is a clique.

Proof. Consider two non-edges {a, b} and {p, q} of T . If the non-edges share an endpoint then they are adjacent in L( T ) 2 since they are already adjacent in L( T ).

Otherwise, since T is a tree, at least one pair of {a, p}, {a, q}, {b, p} and {b, q} is a non-edge in T , otherwise T has a 4-cycle. By definition, {a, b} and {p, q} are adjacent in L( T ) 2 .

If G is the union of two tree-cographs G 1 and G 2 then the maximal cardinality of a clique in L(G) 2 is, simply, the maximum over the clique numbers of L(G 1 ) 2 and L(G 2 ) 2 . The following lemma deals with the join of two tree-cographs. Lemma 3. Let P and Q be tree-cographs and let G be the join of P and Q. Let X be the set of edges that have one endpoint in P and one endpoint in Q. Then (a) X forms a clique in L(G) 2 , (b) every edge of X is adjacent in L(G) 2 to every edge of P and to every edge of Q, and (c) every edge of P is adjacent in L(G) 2 to every edge of Q.

Proof. This is an immediate consequence of the definitions.

⊓ ⊔

For k 3, a k-sun is a graph which consists of a clique with k vertices and an independent set with k vertices. There exist orderings c 1 , . . . , c k and s 1 , . . . , s k of the vertices in the clique and independent set such that each s i is adjacent to c i and to c i+1 for i = 1, . . . , k -1 and such that s k is adjacent to c k and c 1 . A graph is strongly chordal if it is chordal and has no k-sun, for k 3 [8]. Proof. When T is a tree then L(T ) is a blockgraph. Obviously, blockgraphs are strongly chordal. Lubiw proves in [14] that all powers of strongly chordal graphs are strongly chordal.

We strengthen the result of Lemma 4 as follows. Ptolemaic graphs are graphs that are bot

…(Full text truncated)…