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

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