Acta Math Vietnam https://doi.org/10.1007/s40306-018-0273-0 Critical Paired Dominating Sets and Irreducible Decompositions of Powers of Edge Ideals Nguyen Thi Dung 1 · Nguyen Thi Thanh Tam 2 · Hoang Le Truong 3,4 · Hoang Ngoc Yen 5 Dedicated to Professor Le Tuan Hoa on the Occasion of His 60th Birthday Received: 1 September 2017 / Accepted: 6 May 2018 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018 Abstract Let G be a finite simple graph. A set S of vertices is a critical paired dominating set of G, if every vertex is adjacent to a vertex in S and the removal of any vertex does not change the matching number of G. In this paper, we give a characterization of graphs G which has a critical paired dominating set in terms of the irreducible decomposition of powers of the edge ideal associated to G. Keywords Edge ideals · Irreducible ideals · Irreducible decomposition · Matching · Factor-critical graphs · Critical paired dominating set Mathematics Subject Classification (2010) 13A30 · 13D45 · 13E05 · 13H10  Hoang Le Truong hltruong@math.ac.vn; hoang@math.uni-sb.de Nguyen Thi Dung xsdung050764@gmail.com Nguyen Thi Thanh Tam thanhtamnguyenhv@gmail.com Hoang Ngoc Yen hnyen91@gmail.com 1 Thai Nguyen University of Agriculture and Forestry, Thai Nguyen, Vietnam 2 Hung Vuong University, VietTri, Phu Tho, Vietnam 3 Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet Road, 10307 Hanoi, Vietnam 4 Mathematik und Informatik, Universit¨ at des Saarlandes, Campus E2 4, 66123 Saarbr¨ ucken, Germany 5 Thai Nguyen University of Education, Thai Nguyen, Vietnam