MAT-free graphic arrangements and a characterization of strongly chordal graphs by edge-labeling

Ideal subarrangements of a Weyl arrangement are proved to be free by the multiple addition theorem (MAT) due to Abe-Barakat-Cuntz-Hoge-Terao (2016). They form a significant class among Weyl subarrangements that are known to be free so far. The concept of MAT-free arrangements was introduced recently by Cuntz-M{\"u}cksch (2020) to capture a core of the MAT, which enlarges the ideal subarrangements from the perspective of freeness. The aim of this paper is to give a precise characterization of the MAT-freeness in the case of type $A$ Weyl subarrangements (or graphic arrangements). It is known that the ideal and free graphic arrangements correspond to the unit interval and chordal graphs respectively. We prove that a graphic arrangement is MAT-free if and only if the underlying graph is strongly chordal. In particular, it affirmatively answers a question of Cuntz-M{\"u}cksch that MAT-freeness is closed under taking localization in the case of graphic arrangements.


Introduction
A hyperplane arrangement is said to be free if its logarithmic derivation module is a free module [16,11] (cf.Definition 2.1).An important class of free arrangements is that of Weyl arrangements defined by the positive roots of irreducible root systems.There has been considerable interest in finding and characterizing free subarrangements of a Weyl arrangement by combinatorial structures.In the case of type A, Weyl subarrangements are completely determined by graphic arrangements whose freeness is fully characterized by chordal graphs [14,4] (cf.Theorem 2.3).Several certain cases of type B were studied in the connection with signed graphs [4,15,18] yet no complete characterization is known.Arguably, the most fundamental and significant class of Weyl subarrangements known to be free is that of ideal subarrangements derived by the multiple addition theorem (MAT) due to Abe-Barakat-Cuntz-Hoge-Terao [1] (cf.Theorem 2.7).In fact, the MAT applies in a more general setting; based on that Cuntz-Mücksch [2] introduced a new class of free arrangements, the so called MAT-free arrangements (cf.Definition 2.4) which generalizes the ideal subarrangements.
We are especially interested in characterizations of freeness-related properties of type A Weyl subarrangements (= graphic arrangements) in which considerable power and development of graph theory would be brought to bear.It is known that the ideal graphic arrangements are parametrized by unit interval graphs e.g., [19] (cf.Theorem 2.8).Our main result is a complete characterization of MAT-free graphics arrangements by means of strongly chordal graphs (cf.Definition 2.9).We summarize the results in Table 1.We find it interesting that the concept of MAT-freeness which was recently introduced is captured by the notion of strongly chordal graphs which appeared much earlier in literature.This can also be regarded as an analogue of the classical theory of freeness and chordality.We will see in §5.1 that many important concepts in the classical theory such as simplicial vertices and perfect elimination orderings of chordal graphs have their analogous MAT-versions.
Our main result has a number of applications.From the viewpoint of arrangement theory, it gives an affirmative answer to a question of Cuntz-Mücksch in the case of graphic arrangements that MAT-freeness is closed under taking localization (cf.Corollary 2.11).Thanks to the relation {unit interval graphs} {strongly chordal graphs}, it also gives a different and graphical proof that the ideal graphic arrangements are MAT-free (cf.§6(A)).From the viewpoint of graph theory, our main result contributes a new characterization of strongly

Graph class
Weyl subarrangement class Location chordal free [14], [4,Theorem 3.3] strongly chordal MAT-free Theorem 2.10 unit interval ideal e.g., [19,Theorem 16] Table 1: Interplay between graphs and Weyl arrangements in type A. The third row shown in bold indicates the main result of the paper.
chordal graphs via a special type of edge-labelings, which we shall call MAT-labelings (cf.Definition 4.2).
A key ingredient in our proof that strong chordality implies MAT-freeness (the harder part) is a characterization of strongly chordal graphs by their clique intersection posets due to Nevries-Rosenke [10] (cf.Theorem 3.4).Our strategy is to construct an MAT-labeling for a given strong chordal graph with building blocks complete induced subgraphs of the graph.The clique intersection poset of a chordal graph consists of all intersections of maximal cliques of the graph which serves as essential machinery in the construction.We believe that it is worth pursuing further the notion of clique intersection poset for MAT-freeness of larger class of arrangements (see §6(F) for more details).
The remainder of the paper is organized as follows.In §2.1, we recall the definitions and basic facts of free arrangements and chordal graphs.In §2.2, we recall the definitions of MAT-free arrangements and strongly chordal graphs, and give the statement of our main result.In §3, we recall some other useful characterizations of (strongly) chordal graphs.In §4.1, we introduce the notion of MAT-labeling of graphs.In §4.2, we prove the "only if" part of our main Theorem 2.10 (MAT-freeness implies strong chordality).In §5.1, we introduce the notions of MAT-simplicial vertices and MAT-perfect elimination orderings.In §5.2, we prove the "if" part of our main Theorem 2.10 (strong chordality implies MAT-freeness).The proof the main theorem is also included with an example in the end of this section.Finally, in §6, we address some further remarks and suggest some problems for future research.
2 Arrangements, graphs and statement of the main result

Free arrangements
We first review some basic concepts and preliminary results on free arrangements.Our standard reference is [11].Let K be a field, ℓ be a positive integer and V = K ℓ be the ℓ-dimensional vector space over K.A hyperplane in V is a linear subspace of codimension one of V .An arrangement is a finite set of hyperplanes in V .Let A be an arrangement in V .Define the intersection lattice L(A) (of flats) of A by where the partial order is given by reverse inclusion X ≤ Y ⇔ Y ⊆ X for X, Y ∈ L(A).We agree that V is the unique minimal element in L(A) as the intersection over the empty set.Thus L(A) is a geometric lattice which can be equipped with rank function r(X) := codim(X) for X ∈ L(A).We also define rank(A) as the rank of the maximal element where µ denotes the Möbius function µ : L(A) → Z defined recursively by Let {x 1 , . . ., x ℓ } be a basis for the dual space V * and let where Der(S) = {ϕ : S → S | ϕ is K-linear, ϕ(f g) = f ϕ(g) + gϕ(f ) for any f, g ∈ S} is the set of all derivations of S over K.Note that Der(S) is a free S-module with basis {∂/∂x 1 , . . ., ∂/∂x ℓ } consisting of the usual partial derivatives.A non-zero element ϕ = Definition 2.1 (Free arrangements and their exponents [16,11]).An arrangement A is called free with the multiset exp(A) = {d 1 , . . ., d ℓ } of exponents if D(A) is a free S-module with a homogeneous basis {θ 1 , . . ., θ ℓ } such that deg θ i = d i for each i.
Remarkably, when an arrangement is free, the exponents turn out to be the roots of the characteristic polynomial due to Terao.
Theorem 2.2 (Factorization theorem [17], [11,Theorem 4.137 In general it is very hard to characterize the freeness of an arrangement by combinatorial data.It is only possible in some very special class of arrangements, for example, graphic arrangements which we shall recall shortly.Let G be a simple graph (i.e., no loops and no multiple edges) with vertex set V G = {v 1 , . . ., v ℓ } and edge set E G .The graphic arrangement A G in K ℓ is defined by A simple graph is chordal if it does not contain an induced cycle of length greater than three, or C n -free for all n > 3 in shorthand notation.The freeness of graphic arrangements is completely characterized by chordality.
Theorem 2.3 (Freeness and chordality [14], [4,Theorem 3.3]).Let G be a simple graph.The graphic arrangement A G is free if and only if G is chordal.

MAT-free arrangements and the main result
Now we recall the concept of MAT-free arrangements following [2].For X ∈ L(A), we define the localization of A on X by and define the restriction A X of A to X by

(MP2) There does not exist H
An arrangement is called MAT-free if it is empty or admits an MAT-partition.
All irreducible complex reflection arrangements that are MAT-free were characterized in [2].The name "MAT-free arrangement" is made by inspiration of the multiple addition theorem due to Abe-Barakat-Cuntz-Hoge-Terao.
Here d e means d appears e ≥ 0 times in the multiset of exponents.
The addition of {H 1 , . . ., H q } to the arrangement A ′ resulting in A = A ′ ∪ {H 1 , . . ., H q } in Theorem 2.5 is sometimes called an MAT-step [9,Definition 2.11].Thus any MAT-free arrangement is a free arrangement which can be constructed inductively from the empty arrangement by MAT-steps.We save the information for later use (see also [9,Remark 2.15]).
(2) A is free with exp(A) = {d 1 , . . ., d ℓ } ≤ given by the block sizes of the dual partition of π, that is, A remarkable application of the MAT is an affirmative answer for a conjecture of Sommers-Tymoczko [13] on the freeness of ideal subarrangements of Weyl arrangements.
Define the partial order ≥ on Φ + as follows: For an ideal I ⊆ Φ + , the corresponding Weyl subarrangement A I is called the ideal subarrangement.Theorem 2.7 (Ideal MAT-free theorem [1, Theorem 1.1]).Any ideal subarrangement A I is MAT-free, hence free.
In this paper we are mainly interested in the graphic arrangements hence root systems of type A. We will use the following construction of type A root systems.Let {ǫ 1 , . . ., ǫ ℓ } be an orthonormal basis for V , and define Thus one can see that there is a one-to-one correspondence between the graphic arrangements in R ℓ and type A ℓ−1 Weyl subarrangements.
In the case of type A, the ideal subarrangements can be parametrized by unit interval graphs.Recall that a simple graph is a unit interval graph if it is chordal and (claw, net, 3-sun)-free (see Figure 1).Theorem 2.8 (Ideals and unit interval graphs e.g., [19,Theorem 16]).Let G be a simple graph with ℓ vertices.There exists a vertex-labeling of G using elements from [ℓ] so that the graphic arrangement A G is an ideal subarrangement of the Weyl arrangement In the study of interplay between arrangements and graphs it is thus natural to ask which graph class corresponds to the MAT-free graphic arrangements?An answer to this question concerns strongly chordal graphs, a class squeezed between the classes of unit interval and chordal graphs.Definition 2.9 (Strongly chordal graphs e.g., [5]).An n-sun (or trampoline) where we consider u n+1 = u 1 .A simple graph is strongly chordal if it is chordal and n-sun-free for n ≥ 3. See Figure 1 for the n-suns with n = 3, 4.
We are ready to state the main result of the paper (whose proof will be presented in the end of §5.2).

Theorem 2.10 (MAT-freeness and strong chordality). Let G be a simple graph. The graphic arrangement A G is MAT-free if and only if G is strongly chordal.
Cuntz-Mücksch [2, Problem 48] asked if the class of MAT-free arrangements is closed under taking localization.An important consequence of our Theorem 2.10 is an affirmative answer for this question in the case of graphic arrangements.

Corollary 2.11. MAT-freeness of graphic arrangements is closed under taking localization. 3 More on (strongly) chordal graphs
In this section, we recall some other characterizations of (strongly) chordal graphs that will be useful for our discussion later.
First we collect some terminology and notation from graph theory.
The 3-cycle is also called a triangle.The length of a cycle is its number of edges.A chord of C is an edge not in the edge set of C whose endvertices are in the vertex set.
A clique of G is a subset of V G such that every two distinct vertices are adjacent.For each The following characterization of chordal graphs is useful to determine the exponents of the corresponding graphic arrangement (e.g., [4,Lemma 3.4]).

Theorem 3.1 (Chordality and PEO [6]). A simple graph is chordal if and only if it has a perfect elimination ordering.
Let a, b ∈ V G be two distinct vertices which belong to the same connected component of The following characterization of chordality will also be useful for some inductive arguments later.

Theorem 3.2 ([3, Theorem 1]). A simple graph is chordal if and only if every minimal vertex separator is a clique.
A maximal clique is a clique that it is not a subset of any other clique.A largest (or maximum) clique is a clique that has the largest possible number of vertices.Denote by K(G) the set of all maximal cliques of G.
Let G be a chordal graph.Let P G be the poset consisting of all possibly-empty intersections of maximal cliques of G, i.e, where the partial order is given by inclusion We call P G the clique intersection poset 1 of G.Note that P G is a meet-semilattice (not necessarily graded) whose maximal elements are the maximal cliques of G, and minimal element 0 := C∈K(G) C ∈ P G is the clique consisting of the dominating vertices 2 of G.We call an element of P G a node.Remark 3.3.Ho-Lee [7, Lemma 2.1] showed that a nonempty subset S ⊆ V G is a minimal vertex separator if and only if S = C ∩ C ′ for distinct maximal cliques C and C ′ forming an edge in some clique tree of G. Therefore every minimal vertex separator of G belongs to P G .
A k-crown3 (k ≥ 1) is a poset on {x 1 , . . ., x k , y 1 , . . ., y k } with relations x i < y i and x i < y i+1 for all 1 ≤ i ≤ k (counted modulo k) and there are no other relations.See Figure 2 for the k-crowns with k = 3, 4. A poset P is called k-crown-free if there exists no induced subposet 4 of P isomorphic to the k-crown.The following characterization of strongly chordal graphs will play a crucial role in the proof of the "if" part of our main Theorem 2.10 (strong chordality implies MAT-freeness §5.2).

MAT-freeness implies strong chordality 4.1 MAT-labeling of graphs
In this subsection, we show that MAT-free graphic arrangements (Definition 2.4) can be completely determined by a special edge-labeling of graphs.First we show that the condition of being admitted a "partition" of a (nonempty) MAT-free arrangement is actually implied by the three conditions (MP1), (MP2), and (MP3).

Proposition 4.1. An arrangement A is MAT-free if and only if A can be decomposed into a disjoint union of possibly-empty subsets π 1 , . . . , π n of A satisfying (MP1), (MP2), and (MP3).
Proof.If A = ∅, then the statement is clear.Suppose A = ∅.We only need to show the "if" part, namely, the existence of a disjoint union of possibly-empty subsets π 1 , . . ., π n of A satisfying (MP1), (MP2), and (MP3) implies the existence of an MAT-partition of A.
First we show that π 1 = ∅.Suppose to the contrary that π Let p := max{1 ≤ i ≤ n | π i = ∅}.We will show that π k = ∅ for all k ∈ [p] which in turn implies that (π 1 , . . ., π p ) is an MAT-partition of A. Suppose to the contrary that there exists 2 ≤ k < p such that π k = ∅ and choose minimal such k.
. Also, the maximal exponents of B ′ are equal to k − 1.Let q := min{k < i ≤ p | π i = ∅}.Since π q = ∅, we can take H ∈ π q and write B := B ′ ∪ {H}.It is a known fact 6 in the theory of free arrangements that A pair (G, λ) consisting of a simple graph G and a map λ : E G → Z >0 (called (edge-)labeling) is said to be an edge-labeled graph.Now we define a labeling of graphs which characterizes the MAT-freeness of graphic arrangements.Proof.The graphic arrangement A K ℓ (in R ℓ ) is precisely the Weyl arrangement of type A ℓ−1 (also known as the braid arrangement) which has exponents {0, 1, 2, . . ., ℓ − 1}.Corollary 2.6 completes the proof.
Remark 4.5.In fact, K ℓ always has an MAT-labeling λ.We can see this from Theorem 2.7 as K ℓ corresponds to a positive system (in particular, an ideal) of a root system of type A.
It is important to know whether or not a restriction of an MAT-labeling is also an MATlabeling.The proposition below states that it is enough to check the third condition.
Thus the conditions (ML1) and (ML2) are automatically satisfied.Lemma 4.7.Let λ be an MAT-labeling of a simple graph G and Without loss of generality, we may assume e ∈ π F 1 k .Since λ| F 1 is an MAT-labeling, e forms at least k − 1 triangles with edges in E k−1 ∩ (F 1 ∪ F 2 ).Moreover, since λ is an MAT-labeling, e forms at most k − 1 triangles with edges in E k−1 ∩ (F 1 ∪ F 2 ).Therefore e forms exactly k − 1 triangles with edges in

Proof of the implication "MAT-free ⇒ strongly chordal"
In this subsection we prove the "only if" part of our main Theorem 2.10 (MAT-freeness implies strong chordality).
First we need a few preliminary results.Let G = (V G , E G ) be a simple graph.Let χ G (t) be the chromatic polynomial of G (the polynomial that counts the number of proper vertex colorings of G).It is known that χ G (t) = χ A G (t) (e.g., [11,Theorem 2.88]).Let ω(G) denote the clique number, the number of vertices in a largest clique of G. Recall that K(G) denotes the set of all maximal cliques of G. Proposition 4.8.Let G be a chordal graph.Then the following statements hold. 7The height of a positive root β = α∈∆ c α α ∈ Φ + is defined by α∈∆ c α . 8Two edge-labeled graphs (G 1 , λ 1 ) and (G 2 , λ 2 ) are isomorphic if there exists a bijection σ : G, then we say that two labelings λ 1 and λ 2 are the same (or isomorphic) if (G, λ 1 ) and (G, λ 2 ) are isomorphic.
(1) The maximal exponents of A G are equal to ω(G)−1.In addition, the number of maximal exponents of A G equals the number of largest cliques of G.
(2) If λ is an MAT-labeling of G, then the endvertices of each e ∈ π n where n = ω(G) − 1 are contained in a unique maximal clique of G. Furthermore, the map φ : π n → K(G) defined by φ(e) = the maximal clique containing the endvertices of e induces a bijection π n ≃ φ(π n ) and φ(π n ) = {all largest cliques of G}.
The assertions clearly hold.We may assume that ℓ ≥ 3 and G is not complete.We proceed by induction on

g., [12, Theorem 3]).
Let m 1 and m 2 denote the numbers of cliques consisting of ω(G) many vertices in G 1 and G 2 , respectively.Since there is no clique of G containing both vertices in A and in B, the number of largest cliques of G equals m 1 + m 2 .By the induction hypothesis, the chromatic polynomials of G 1 and G 2 can be expressed as , where f (t), g(t) ∈ Z[t] are the products of some linear factors with roots strictly smaller than ω Let Thus every largest clique of G contains the endvertices of at least one edge in π n .Hence every largest clique of G belongs to φ(π n ) and |φ(π n )| ≥ |π n |.Thus φ(π n ) is precisely the set consisting of the largest cliques.This completes the proof.
In general, a restriction of an MAT-labeling is not an MAT-labeling (e.g., when we restrict to any edge in π k with k > 1).We show below that it is the case for restriction to certain subset.Recall that P G denotes the clique intersection poset of a chordal graph G ( §3).

Lemma 4.9. If λ is an MAT-labeling of a chordal graph G, then the restriction λ| E G[X] is an MAT-labeling of the subgraph (V G , E G[X]
) for any node X ∈ P G .
Proof.We proceed by induction on n = ω(G) − 1. Again it is easily seen that the assertion holds true for n = 0, 1.Now suppose n ≥ 2. First we consider the case X = C where C is a largest clique of G.Note that by Proposition 4.6, it suffices to prove that λ| E G[C] satisfies (ML3).
Let e = {u, v} ∈ π n be the unique edge in π n whose endvertices are contained in C (Proposition 4.8).The clique C is the union of two largest cliques C ′ = C \ {v} and are MAT-labelings by the induction hypothesis.
To show (ML3) of λ| E G[C] , it suffices to prove that every edge in π C k forms at least k − 1 triangles with edges in ∈ X.Thus either X ⊆ C ′′ or X ⊆ C ′ .Note also that every maximal clique of G which is not largest is a maximal clique of G ′ .Therefore the node X is the intersection of some maximal cliques of G ′ .Hence X ∈ P G ′ and by the induction hypothesis, λ| We are ready to prove the main result of this subsection.

Theorem 4.10 (MAT-freeness implies strong chordality). If a simple graph G admits an MAT-labeling, then G is strongly chordal.
Proof.Suppose that G is not strongly chordal.Note that G is chordal by Proposition 4.3.Then G contains an n-sun S n (Definition 2.9) as an induced subgraph for some n ≥ 3.
Let Z := {u 1 , . . ., u n } be the central clique of S n , T i := {u i , v i , u i+1 } the vertex set of the triangle around Z, and . By Lemmas 4.9 and 4.7, G 0 admits an MAT-labeling. Suppose . Therefore both u i and u j are adjacent to v k .This implies {u i , u j , v k } = T k and hence j = i ± 1. Thus the cycle in G 0 consisting of edges {u 1 , u 2 }, . . ., {u n−1 , u n }, {u n , u 1 } has no chords and its length is four or more.Therefore G 0 is not chordal, which is a contradiction.Now we consider n = 3.

Strong chordality implies MAT-freeness 5.1 MAT-simplicial vertices and MAT-perfect elimination orderings
The proof of the "if" part of our main Theorem 2.10 (strong chordality implies MATfreeness) requires more effort.We need a deeper understanding of the structure of graphs having MAT-labelings.In this subsecion, we develop a fundamental study on such graphs analogous to the theory of (strongly) chordal graphs by introducing MAT-versions of simplicial vertex and perfect elimination ordering.

Definition 5.1 (MAT-simplicial vertices). Given an edge
Next we show the existence of MAT-simplicial vertices in the graphs having MAT-labelings.
Lemma 5.2.Let (G, λ) be an edge-labeled graph such that |V G | ≥ 2 and λ is an MAT-labeling of G.
(1) If G = K ℓ is a complete graph, then the endvertices of the edge with maximal label are MAT-simplicial.
Proof.First we prove part (1).Let e 0 = {u 0 , v 0 } ∈ E G be the edge with maximal label.It suffices to prove that v 0 is MAT-simplicial.First, (MS1) is clear.Next we show (MS2).Note that λ| E G\e 0 is an MAT-labeling.Then by Lemma 4.9, the labelings λ| are MAT-labelings, where Lastly, we show (MS3).Let u, v ∈ X and write λ({u, v}) = k.We want to show k < max{λ({u, , since λ| E G\v 0 is also an MAT-labeling by Lemma 4.9.Therefore (MS3) holds.Thus v 0 is MAT-simplicial.Now we prove part (2).We proceed induction on ℓ = |V G |.If ℓ = 2, then the assertion holds trivially.Suppose ℓ ≥ 3. We may assume that G is connected.Let a, b ∈ V G be nonadjacent vertices and S a minimal (a, b)-separator.Then S is a clique by Theorem 3.2 and S ∈ P G by Remark 3.3.Hence λ| E G[S] is an MAT-labeling of G[S] by Lemma 4.9.
Let A be the vertex set of the connected component containing a of G \ S and B := . Therefore e forms exactly k − 1 triangles with edges in . Suppose that at least one endvertex of e belongs to A. Then e cannot form a triangle with a vertex in B. Hence e forms exactly k − 1 triangles with edges in Similarly, B \ S contains an MAT-simplicial vertex of G. Therefore G has two nonadjacent MAT-simplicial vertices.
The following is a first important property of MAT-simplicial vertices.Proposition 5.3.Let (G, λ) be an edge-labeled graph with |V G | ≥ 2. Suppose that v ∈ V G is an MAT-simplicial vertex of (G, λ).The following are equivalent.
In particular, any MAT-PEO is a PEO.The theorem below exhibits a strong connection between MAT-labelings and MAT-PEOs which can be seen as an analogue of Theorem 3.1.
We complete this subsection by giving two lemmas on (extensions of) MAT-labelings and MAT-PEOs of complete graphs.MAT-labelings of complete graphs will play a crucial role in the next subsection.
Lemma 5.6.Let G = K ℓ be a complete graph and W ⊆ V G .
(1) Let λ be an MAT-labeling of G.
When r = 0, P 0 G = { 0}.Since G[ 0] is a complete graph (or null graph), there exists an MAT-labeling of G[ 0] by Lemma 5.6.Now suppose r > 0. Then by the induction hypothesis there exists a family We prove the following claim.
By Lemma 5.7, there exists an MAT-labeling Now we return to the proof of Lemma 5.10.Let T be the set consisting of all nodes covered by X.Then use Lemma 5.6 (2) We are ready to prove the main result of this subsection.Theorem 5.12 (Strong chordality implies MAT-freeness).If G is a strongly chordal graph, then G admits an MAT-labeling.Proof.By Lemma 5.10, there exists a family {λ Considering the antichain K(G) of P G consisting of the maximal cliques of G, we can construct an MAT-labeling λ of G by using Lemma 5.9 and the "gluing trick" (Theorem 5.8).More precisely, we show that for any T ⊆ K(G), there exists an MAT-labeling λ Finally we present the proofs of the main result of the paper and its corollary.
Proof of Corollary 2.11.Taking localization on a flat of a graphic arrangement is equivalent to taking an induced subgraph of the underlying graph.The proof follows from Theorem 2.10 and a simple fact that the class of strongly chordal graphs is closed under taking induced subgraphs.
We close this section by giving an example to illustrate the construction in Theorem 5.12.
Example 5.13.Let G be a unit interval graph in Figure 3.Its clique intersection poset P G is given in Figure 4. First we need to find a family F (P G ) = {λ X } X∈P G consisting of MAT-labelings one for each G[X] such that F (P G ) is closed under inclusion mentioned in Lemma 5.10.This can be done inductively from the bottom to top starting from the minimum element 0. For example, to find a desired MAT-labeling λ 3 ∈ F (P G ) of G[X] where X = {v 2 , v 3 , v 4 , v 5 } provided that the compatible MAT-labelings of G[Y ] for all Y 's covered by X (in this case {v 4 , v 5 } and {v 2 , v 3 , v 4 }) were given, we use Lemma 5.7 (and Lemma 5.6(2) if ∪Y X).Combining the resulting MAT-labelings λ i ∈ F (P G ) (1 ≤ i ≤ 4) of the maximal cliques by the "gluing trick" (Theorem 5.8) yields an MAT-labeling of G. Figure 5 shows a gluing ((λ 1 ∪ λ 2 ) ∪ λ 3 ) ∪ λ 4 and how the exponents change in each inductive step, which we call an "exponent growth process".Note that although MAT-labeling of G is uniquely determined by λ i 's, gluing order is not necessarily unique.For example, the gluing λ 1 ∪ (λ 2 ∪ (λ 3 ∪ λ 4 )) derived from the same method gives the same output but different exponent growth process: Figure 3: A unit interval (hence strongly chordal) graph G on 7 vertices with an MATlabeling constructed by using Theorem 5.12.The corresponding graphic arrangement A G is free with exponents {0, 1, 2, 2, 2, 3, 3}.

Further remarks and open problems
In this section we address some remarks and suggest problems for future research.
(A) As noted in Introduction, our Theorem 2.10 gives an alternative proof that the ideal graphic arrangements are MAT-free (type A of Theorem 2.7).We give here two examples to illustrate the difference between two methods.The original proof of the ideal MATfree theorem is inductive on the height of ideals [1, §5], and in each inductive MAT-step only some of maximal exponents get increased by 1.This yields a rigorous exponent growth process hence differs from our construction in Theorem 5.12.For example, the unit interval graph G in Figure 3 with the given vertex-labeling has its corresponding graphic arrangement A G an ideal subarrangement of the Weyl arrangement A Φ + (A 6 ) .The exponent growth process following the ideal MAT-free theorem is given in Figure Figure 5: An exponent growth process for the graph in Figure 3 following the "gluing trick" in Theorem 5.12.
6 which differs from that in Figure 5.Our construction applies also to strongly chordal graphs that are not unit interval graphs.Another way to see the difference between two methods is to consider MAT-labelings of complete graphs, see Remark 4.5, Lemma 5.6(2) and Figure 7.
Figure 6: Exponent growth process for the graph in Figure 3 following the ideal MAT-free theorem.(B) Cuntz-Mücksch [2,Example 22] showed that MAT-freeness is in general not closed under taking restriction.Their example is a non-MAT-free restriction to a hyperplane of the Weyl arrangement of type E 6 .We give here a different example (with a smaller number of hyperplanes) thanks to the fact that the class of strongly chordal graph is not closed under taking edge-contraction.Consider the rising sun (which is a strongly chordal graph) with its edge e displayed in Figure 8. Taking the contraction of e results in the 3-sun which is not strongly chordal.(C) Strongly chordal graphs are the intersection graphs of unit balls in R-trees [8].Therefore they can be considered as generalization of unit interval graphs in the perspective of intersection graphs.
(D) Strongly chordal graphs are also known as the graphs having a strong perfect elimination ordering (SPEO) [5], i.e., a PEO (v 1 , . . ., v ℓ ) with the property that for all i < j, k < q if {v i , v k }, {v i , v q }, {v j , v k } are edges, then {v j , v q } is an edge.It would be interesting to find a (more direct) connection between SPEO and MAT-PEO.
(E) If an arrangement A is MAT-free, then A is accurate [9, Theorem 1.2] i.e., A is free with exp(A) = {d 1 , . . ., d ℓ } ≤ and there exists for each 0 ≤ p < ℓ a p-codimensional flat X ∈ L(A) such that A X is free with exp(A X ) = {d 1 , . . ., d ℓ−p } ≤ .Characterize the accuracy of graphic arrangements.We are able to show that if G is an n-sun, then A G is accurate (but not MAT-free).
(F) From Theorems 2.10 and 3.4, we now know that the MAT-freeness of graphic arrangements can be characterized by a poset structure, the clique intersection poset of chordal graphs.Define a "clique intersection poset" of an arbitrary (supersolvable) arrangement and characterize the MAT-freeness of the arrangement by the poset.It is related to another question of Cuntz-Mücksch [2, Problem 47] which asked if the MAT-freeness can be characterized by a partial order on the hyperplanes, generalizing the classical partial order ( §2.2) on the positive roots of an irreducible root system.

Figure 1 :
Figure 1: Some obstructions to unit interval and strongly chordal graphs.

1 .Proposition 4 . 4 .6
Here cl(F ) for F ⊆ E G denotes the closure of F in the matroid sense.Namely, an edge e ∈ E G is in cl(F ) when the two endvertices of e are connected by edges in F .(ML3) Every e ∈ π k forms exactly k − 1 triangles (3-cycles) with edges in E k−1 .Proposition 4.3.Let G be a simple graph.The graphic arrangement A G is MAT-free if and only if G admits an MAT-labeling.Proof.This is a translation of Definition 2.4 into graphical terms with the use of Proposition 4.Thus characterizing the MAT-freeness of graphic arrangements amounts to characterizing the graphs having MAT-labelings.Here are first and simple facts on MAT-labelings.Denote by K ℓ the complete graph on ℓ vertices.If λ is an MAT-labeling of K ℓ , then |π k | = ℓ − k for all k ∈ [ℓ − 1].Here is the precise statement: "Let A be an arrangement and let H ∈ A. If A ′ := A \ {H} is free with maximal exponent m, then |A ′ | − |A H | ≤ m."We believe that this fact is well known among experts, but we give here a short proof for the sake of completeness.There exists a polynomial B such that deg B = |A ′ | − |A H | and D(A ′ )α H is contained in the ideal (α H , B) ⊆ S [11, Lemma 4.39 and Proposition 4.41].If deg B > m, then D(A ′ )α H ⊆ (α H ), hence D(A ′ ) = D(A).Therefore A is free and exp(A) = exp(A ′ ).This implies |A ′ | = |A|, a contradiction.Thus |A ′ | − |A H | = deg B ≤ m.

Proposition 4 . 6 .
Let λ be an MAT-labeling of a simple graph G and F ⊆ E G .Then the restriction λ| F is an MAT-labeling of the subgraph 3), the maximal exponents of G are equal to ω(G) − 1 and the number of maximal exponents of G is m 1 + m 2 .Now we prove part (2).If n = 0, the assertions hold trivially.If n = 1, then G is a forest and λ is a constant labeling whose value is 1.Hence the assertions also hold.Now suppose n ≥ 2. Let e ∈ π n .Clearly, there exists C ∈ K(G) such that e ∈ G[C].Suppose that there exist two distinct C 1 , C 2 ∈ K(G) such that e ∈ G[C 1 ∩ C 2 ].Since λ E G\e is an MAT-labeling, A G\e is free and hence G \ e is chordal.By the maximality of C 1 , C 2 , there exist u ∈ C 1 \ C 2 and v ∈ C 2 \ C 1 such that {u, v} / ∈ E G .Then u, v and the endvertices of e form a 4-cycle which is chordless in G \ e, a contradiction.Thus the endvertices of each edge in π n are contained in exactly one maximal clique of G. Therefore the map φ is well-defined.Moreover, |φ(π n )| ≤ |π n |.
we can find an edge e ∈ E G[X] ∩ π n .Let C denote the largest clique such that e ∈ E G[C] .By the definition of X, there exists a maximal clique D of G such that X ⊆ D = C. Thus e ∈ E G[C∩D] .Take a vertex c ∈ C \ D. By the maximality of D, there exists d ∈ D such that {c, d} / ∈ E G .Then we obtain a chordless 4-cycle in G ′ = G \ π n formed by c, d and the endvertices of e, which contradicts to the chordality of G ′ .Thus E G[X] ∩ π n = ∅.Let C be a largest clique of G and e = {u, v} ∈ π n be the unique edge in π n whose endvertices are contained in C. Then C = C ′ ∪ C ′′ where C ′ = C \ {v} and C ′′ = C \ {u} are largest cliques of G ′ = G \ π n .By the above discussion, e / ∈ E G[X] hence either u / ∈ X or v / is a complete graph, then the endvertices of the edge e 0 in G[A ∪ S] with maximal label is MAT-simplicial in G[A ∪ S] by part (1).Since λ| E G[S] is an MAT-labeling, at least one endvertex of e 0 belongs to A \ S by Proposition 4.4, which is a desired MAT-simplicial vertex.If G[A ∪ S] is not a complete graph, then by the induction hypothesis G[A ∪ S] has two nonadjacent MAT-simplicial vertices.At least one of them belongs to A \ S since S is a clique.Thus A \ S contains an MAT-simplicial vertex of G.
To show (ML2), suppose cl G (π k ) ∩ E k−1 = ∅ and take e ∈ cl G (π k ) ∩ E k−1 .Then there exists a cycle C in G such that e ∈ C andC \ e ⊆ π k .If e is not incident to v (in particular, v is not a vertex of C), then e ∈ E ′ k−1 and C \ e ⊆ π ′ k .Therefore e ∈ cl G\v (π ′ k ) ∩ E ′ k−1 = ∅, a contradiction.Hence e is incident to v, and C contains an edge {v, w} with λ({v, w}) = k.Write e = {u, v}.Then {u, w} ∈ E G by (MS1) and λ({u, w}) < max{λ({u, v}), λ({v, w})} = k by (MS3).Hence {u, w} ∈ E ′ k−1 (in particular, C has length at least 4).Moreover, {u, w} and C \ {{v, w}, e} ⊆ π ′ k form a cycle and hence {u, w}∈ cl G\v (π ′ k ) ∩ E ′ k−1 = ∅, a contradiction.Finally, we prove (ML3).Let e ∈ π k .If e ∈ π ′ k (i.e, e is not incident to v), then e forms exactly k − 1 triangles with some edges in E ′ k−1 .If one endvertex of e is not adjacent to v, then e and v cannot form a triangle.If both endvertices of e are adjacent to v, then at least one edge of the triangle containing e and v has label greater than k by (MS3).In either case, e forms exactly k − 1 triangles with some edges inE k−1 .Now consider e ∈ π k \ π ′ k (i.e., e is incident to v).By (MS2), we can index the elements of N G (v) as {u 1 , . . ., u d } where k ≤ d = deg G (v) so that e = {u k , v} and λ({u i , v}) = i for every i ∈ [d].Hence e forms exactly k − 1 triangles given by {u i , u k , v} for i ∈ [k − 1] with some edges in E k−1 .Thus λ is an MAT-labeling.Definition 5.4 (MAT-PEO).Given an edge-labeled graph (G, λ), an ordering (v 1 , . .

Claim 5 . 11 . 1 Gand T ⊆ P r− 1 G
Let X ∈ P r G \ P r−a set consisting of some nodes covered by X.Then there exists an MAT-labelingλ T of G[∪ Y ∈T Y ] satisfying λ T | E G[Y ] = λ Y for any Y ∈ T .Proof ofClaim 5.11.We prove by induction on |T |.If |T | = 1, then it is clear.Suppose |T | ≥ 2. By Lemma 5.9, there exist for every Y ∈ T .This can be done by induction on |T | very similar to the proof of Claim 5.11.Then take T = K(G).
By Proposition 4.6, we only need to prove (ML3) of λ| E G\v .Let e ∈ π ′ k .Since e ∈ π ′ k ⊆ π k , e forms exactly k − 1 triangles with edges in E k−1 .These triangles do not contain the vertex v because by (MS3) the number of edges incident to v with label less than k is at most 1.Therefore e forms exactly k − 1 triangles with edges inE ′ k−1 .Thus λ| E G\v is an MAT-labeling of G \ v.Next we prove (2) ⇒ (1).Let k ∈ Z >0 .By (MS2), v is a leaf or an isolated vertex of π k .Moreover, since π ′ k is a forest, π k is a forest.This shows (ML1).