Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)   Publisher：Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Gora  

DOI： 10.7151/dmgt.2231

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Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media {LLC}  

DOI： 10.1007/s00373-020-02152-1

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Other Link： http://link.springer.com/article/10.1007/s00373-020-02152-1/fulltext.html

Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

DOI： 10.1007/s00373-020-02137-0

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Other Link： http://link.springer.com/article/10.1007/s00373-020-02137-0/fulltext.html

Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media {LLC}  

DOI： 10.1007/s00373-018-2001-x

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Language：English   Publishing type：Research paper (scientific journal)  

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Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.1007/s00373-018-1932-6

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Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.1002/jgt.22255

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Language：English   Publishing type：Research paper (scientific journal)  

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：American Mathematical Society  

In this paper, we show that two balanced triangulations of a closed surface are not necessarily connected by a sequence of balanced stellar subdivisions and welds. This answers a question posed by Izmestiev, Klee and Novik. We also show that two balanced triangulations of a closed surface are connected by a sequence of three local operations, which we call the pentagon contraction, the balanced edge subdivision and the balanced edge weld. In addition, we prove that two balanced triangulations of the 2-sphere are connected by a sequence of pentagon contractions and their inverses if none of them are the octahedral sphere.

DOI： 10.1090/proc/13775

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

We shall determine exactly two (P, Q)-irreducible even triangulations of the projective plane. This result is a new generating theorem of even triangulations of the projective plane, that is, every even triangulation of the projective plane can be obtained from one of those two (P, Q)-irreducible even triangulations by a sequence of two expansions called a P-expansion and a Q-expansion, which were used in Batagelj (1984, 1989), Drapal and Lisonek (2010). Furthermore, we prove that for any closed surface F-2 there are finitely many (P, Q)-irreducible even triangulations of F-2. (C) 2017 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.disc.2017.06.018

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

We discuss the existence of minors of given graphs in optimal 1-planar graphs. As our first main result, we prove that for any graph H, there exists an optimal 1-planar graph which contains H as a topological minor. Next, we consider minors of complete graphs. It is easily obtained from Mader's result (Mader, 1968) that every optimal 1-planar graph has a K-6-minor. In the paper, we characterize optimal 1-planar graphs having no K-7-minor. (C) 2017 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.disc.2017.01.022

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：UP FAMNIT  

DOI： 10.26493/1855-3974.552.bf3

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：UNIV ZIELONA GORA  

The looseness of a triangulation G on a closed surface F-2, denoted by xi(G), is defined as the minimum number k such that for any surjection c : V(G) -> {1, 2, ..., k + 3}, there is a face uvw of G with c(u), c(v) and c(w) all distinct. We shall bound xi(G) for triangulations G on closed surfaces by the independence number of G denoted by a(G). In particular, for a triangulation G on the sphere, we havexi(G) <= 11 alpha(G) - 10/6and this bound is sharp. For a triangulation G on a non-spherical surface F-2, we havexi(G) <= 2 alpha(G)+ l(F-2) - 2,where l(F-2) = left perpendicular(2 - chi(F-2))/2right perpendicular with Euler characteristic chi(F-2).

DOI： 10.7151/dmgt.1870

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER JAPAN KK  

In this paper, we examine relationship of graphs on surfaces: triangulations, quadrangulations and optimal -planar graphs. For a given quadrangulation of a closed surface can be extended to a triangulation by adding a diagonal edge in every face of . We show that every quadrangulation of with at least six vertices can be extended to a -connected triangulation. Moreover, we show that every -connected triangulation of has a -connected spanning quadrangulation subgraph. As corollaries of these results, we show that every optimal -planar graph has a -connected triangulation subgraph, and that every plane -connected triangulation can be extended to an optimal -planar graph by adding some edges.

DOI： 10.1007/s00373-015-1568-8

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER  

A graph is 1-planar if it can be embedded in the plane with at most one crossing per edge. It is known that the problem of testing 1-planarity of a graph is NP-complete. In this paper, we study outer-1-planar graphs. A graph is outer-1-planar if it has an embedding in which every vertex is on the outer face and each edge has at most one crossing. We present a linear time algorithm to test whether a given graph is outer-1-planar. The algorithm can be used to produce an outer-1-planar embedding in linear time if it exists.

DOI： 10.1007/s00453-014-9890-8

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：CHARLES BABBAGE RES CTR  

Let G be a simple quadrangulation on a closed surface F-2. A face-contraction and a 4-cycle removal are two reductions for quadrangulations defined in this paper.
G is irreducible if any face-contraction breaks the simplicity of G,
G is D-3-irreducible if G has minimum degree at least 3 and any face-contraction or any 4-cycle removal either breaks the simplicity or reduces the minimum degree to less than 3,
G is K-3-irreducible if G is 3-connected and any face-contraction or any 4-cycle removal breaks the simplicity or the 3-connectedness of the graph,
G is S-4-irreducible if G has no separating 4-cycle and any face-contraction breaks the simplicity or creates a separating 4-cycle.
In [7], it was shown that except the sphere and the projective plane, the irreducibility and the D-3-irreducibility of quadrangulations are equivalent. In this paper, we shall prove that for all surfaces, the D-3-irreducibility and the K-3-irreducibility of quadrangulations are equivalent. We also prove that for the sphere, the projective plane and the torus, the D-3-irreducibility and the S-4-irreducibility of quadrangulations are equivalent, but this does not hold for surfaces of high genus.

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Language：English   Publishing type：Research paper (scientific journal)  

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

A 1-planar graph is a graph that can be embedded in the plane with at most one crossing per edge. It is known that testing 1-planarity of a graph is NP-complete.
In this paper, we consider maximal 1-planar graphs. A graph G is maximal 1-planar if addition of any edge destroys 1-planarity of G. We first study combinatorial properties of maximal 1-planar embeddings. In particular, we show that in a maximal 1-planar embedding, the graph induced by the non-crossing edges is spanning and biconnected.
Using the properties, we show that the problem of testing maximal 1-planarity of a graph G can be solved in linear time, if a rotation system Phi (i.e., the circular ordering of edges for each vertex) is given. We also prove that there is at most one maximal 1-planar embedding xi of G that is consistent with the given rotation system Phi. Our algorithm also produces such an embedding in linear time, if it exists. (C) 2013 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.tcs.2013.09.029

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Language：English   Publishing type：Research paper (international conference proceedings)  

A graph is 1-planar if it can be embedded in the plane with at most one crossing per edge. A graph is outer-1-planar if it has an embedding in which every vertex is on the outer face and each edge has at most one crossing. We present a linear time algorithm to test whether a graph is outer-1-planar. The algorithm can be used to produce an outer-1-planar embedding in linear time if it exists. © 2013 Springer International Publishing Switzerland.

DOI： 10.1007/978-3-319-03841-4-7

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Language：English   Publishing type：Research paper (international conference proceedings)  

A 1-planar graph is a graph that can be embedded in the plane with at most one crossing per edge. It is known that testing 1-planarity of a graph is NP-complete. A 1-planar embedding of a graph G is maximal if no edge can be added without violating the 1-planarity of G. In this paper we show that the problem of testing maximal 1-planarity of a graph G can be solved in linear time, if a rotation system (i.e., the circular ordering of edges for each vertex) is given. We also prove that there is at most one maximal 1-planar embedding of G that preserves the given rotation system, and our algorithm produces such an embedding in linear time, if it exists. © 2013 Springer-Verlag.

DOI： 10.1007/978-3-642-36763-2_30

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER JAPAN KK  

We show that an Eulerian triangulation of the Klein bottle has chromatic number equal to six if and only if it contains a complete graph of order six, and it is 5-colorable, otherwise. As a consequence of our proof, we derive that every Eulerian triangulation of the Klein bottle with face-width at least four is 5-colorable.

DOI： 10.1007/s00373-011-1063-9

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WILEY-BLACKWELL  

Let G be a quadrangulation on a surface, and let f be a face bounded by a 4-cycle abcd. A face-contraction of f is to identify a and c (or b and d) to eliminate f. We say that a simple quadrangulation G on the surface is k-minimal if the length of a shortest essential cycle is k(>= 3), but any face-contraction in G breaks this property or the simplicity of the graph. In this article, we shall prove that for any fixed integer k >= 3, any two k-minimal quadrangulations on the projective plane can be transformed into each other by a sequence of Y-rotations of vertices of degree 3, where a Y-rotation of a vertex v of degree 3 is to remove three edges vv(1), vv(3), vv(5) in the hexagonal region consisting of three quadrilateral faces vv(1)v(2)v(3), vv(3)v(4)v(5), and vv(5)v(6)v(1), and to add three edges vv(2), vv(4), vv(6). Actually, every k-minimal quadrangulation (k >= 4) can be reduced to a (k-1)-minimal quadrangulation by the operation called Mobius contraction, which is mentioned in Lemma 13. (c) 2011 Wiley Periodicals, Inc. J Graph Theory 69: 301313, 2012

DOI： 10.1002/jgt.20583

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：UNIV ZIELONA GORA  

A graph is called 1-planar if there exists a drawing in the plane so that each edge contains at most one crossing. We study maximal 1-planar graphs from the point of view of properties of their diagrams, local structure and hamiltonicity.

DOI： 10.7151/dmgt.1639

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WILEY-BLACKWELL  

We identify the structures of 4-connected projective-planar graphs which generate their inequivalent embeddings on the projective plane, showing two series of graphs the number of whose inequivalent embeddings is held by O(n) with respect to the number of its vertices n. (C) 2010 Wiley Periodicals, Inc. J Graph Theory 68: 213-228, 2011

DOI： 10.1002/jgt.20553

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

A graph with at least 2k + 2 vertices is said to be k-extendable if any independent set of k edges in it extends to a perfect matching. We shall show that every 5-connected graph G of even order embedded on a closed surface F(2), except the sphere, is 2-extendable if rho(G) &gt;= 7 - 2x (F(2)), where rho(G) stands for the representativity of G on F(2) and x (F(2)) for the Euler characteristic of F(2). (C) 2011 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jctb.2011.02.001

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

A graph is said to be k-extendable if any independent set of k edges extends to a perfect matching. in this paper, we shall characterize the forbidden structures for 5-connected graphs on the Klein bottle to be 2-extendable. This fact also gives us a sharp lower bound of representativity of 5-connected graphs embedded on the Klein bottle to have such a property, which was considered in Kawarabayashi et al. (submitted for publication) [4]. (C) 2010 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.disc.2010.06.020

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER TOKYO  

A graph is said to be k-extendable if any independent set of k edges extends to a perfect matching. We shall show that every 5-connected graph of even order embedded on the projective plane and every 6-connected one embedded on the torus and the Klein bottle is 2-extendable and characterize the forbidden structures for 5-connected toroidal graphs to be 2-extendable.

DOI： 10.1007/s00373-010-0927-8

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER TOKYO  

In this paper, we shall give a constructive characterization of triangulations on the nonorientable surface of genus 3 without K (6)-minors. Our characterization implies that every 5-connected triangulation and every 4-representative triangulation on the surface has a K (6)-minor.

DOI： 10.1007/s00373-010-0931-z

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

We show that, for any given non-spherical orientable closed surface F(2), there exists an optimal 1-planar graph which can be embedded on F(2) as a triangulation. On the other hand, we prove that there does not exist any such graph for the nonorientable closed surfaces of genus at most 3. (C) 2009 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.disc.2009.07.016

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Yokohama City Unicersity and Yokohama National University  

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Yokohama City Unicersity and Yokohama National University  

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SIAM PUBLICATIONS  

In this paper, we examine the re-embeddability of maximum 1-planar graphs. In particular, we prove that every optimal 1-planar graph is uniquely 1-embeddable on the sphere except for a sequence of graphs that are minimal with respect to certain reductions. These optimal 1-planar graphs are closely related to their quadrangular subgraphs. We also give a generating theorem for optimal 1-planar graphs.

DOI： 10.1137/090746835

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Language：English   Publishing type：Research paper (scientific journal)  

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

In this paper, we show that any two even triangulations Oil the same closed Surface with the same and sufficiently large number of vertices can be transformed into each other by a sequence of two specifically defined deformations called an N-flip and a P(2)-fliP, up to homeomorphism, if they have the same homological structure. (C) 2008 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jctb.2008.06.006

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

We shall show that any two even triangulations G and G' on the projective plane with vertical bar V(G)vertical bar = vertical bar V(G')vertical bar >= 14 can be transformed into each other by two operations called an N-flip and P-2-flip, if and only if both of them are simultaneously 3-colorable, or not. (C) 2007 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.disc.2007.10.015

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Language：English   Publishing type：Research paper (scientific journal)  

In this paper, we show that any two even triangulations on the same closed surface with the same and sufficiently large number of vertices can be transformed into each other by a sequence of two specifically defined deformations called an N-flip and a P2-flip, up to homeomorphism, if they have the same homological structure. © 2008 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.endm.2008.06.020

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：JOHN WILEY & SONS INC  

We shall determine the 20 families of irreducible even triangulations of the projective plane. Every even triangulation of the projective plane can be obtained from one of them by a sequence of even-splittings and attaching octahedra, both of which were first given by Batagelj [2]. (c) 2007 Wiley Periodicals, Inc.

DOI： 10.1002/jgt.20269

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

We show that for any closed surface F with chi(F) &lt;= -4 (or chi(F) &lt;= -2), there exist graphs that triangulate the torus or the Klein bottle (or the projective plane) and that quadrangulate F. We also give a sufficient condition for a graph triangulating a closed surface to quadrangulate some other surface. (c) 2006 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jctb.2006.05.005

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：JOHN WILEY & SONS INC  

A triangulation is said to be even if each vertex has even degree. For even triangulations, define the N-flip and the P-2-flip as two deformations preserving the number of vertices. We shall prove that any two even triangulations on the sphere with the same number of vertices can be transformed into each other by a sequence of N- and P-2-flips. (c) 2005 Wiley Periodicals, Inc.

DOI： 10.1002/jgt.20132

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

It has already been proved that given two closed surfaces F-1(2) and F-2(2) with 2 chi(F-1(2))-chi((2)(F)2) &gt;= 4, there exists a triangulation on F-1(2) which can be embedded on F-2(2) as a quadrangulation. In this paper we refine that result, showing that there exists an integer go such that for any two closed surfaces with genus g(1) &gt;= g(0) and genus g(2) satisfying 2 chi(F-1(2))-chi(F-2(2)) &gt;= O(g(1)), there exists a triangulation of the first surface which can be re-embedded on the second as a quadrangulation. Moreover, on the right-hand side of the inequality, we obtain a concrete expression which is asymptotically O (g(1)). We also obtain similar results for non-orientable surfaces. (c) 2005 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.disc.2004.12.025

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Language：English   Publishing type：Thesis (other)  

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Yokohama City University and Yokohama National University  

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Venue：Hatoba Hall(Yokohama), Japan  

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Venue：Sendai International Center, Japan  

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Venue：東京大学駒場キャンパス  

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Venue：横浜国立大学 みなとみらいキャンパス  

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Venue：熊本大学  

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Venue：Grand Hotel Permon, Podbansk\'{e}, Slovakia  

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Venue：横浜国立大学  

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Venue：Holiday Inn Resort Los Cabos, Mexico  

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Venue：情報科学専門学校（岩崎学園）  

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Venue：新潟大学  

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Venue：Toki Messe, Niigata  

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Venue：熊本大学工学部百周年記念館  

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Venue：Ramada Resort (Kranjska Gora, Slovenia)  

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Venue：学習院大学  

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Venue：Yokohama National University  

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Venue：Smolenice castle, Slovakia  

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Venue：学習院大学  

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Venue：九州大学  

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Venue：加計国際学術交流センター  

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Venue：奈良県文化会館  

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Venue：横浜国立大学  

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Venue：Hotel Tatry, Poland  

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Venue：横浜国立大学  

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Venue：茨城大学工学部日立キャンパス  

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Venue：東海大学教育開発研究所  

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Venue：琉球大学研究者交流会館  

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Venue：龍谷大学瀬田キャンパス  

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Venue：Yokohama National University  

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Venue：横浜国立大学  

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Venue：京都大学数理解析研究所  

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Venue：東京電気通信大学  

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Venue：横浜国立大学  

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Venue：\'{E}cole Normale Su\'{p}erieure, Paris  

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Venue：横浜国立大学  

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Venue：Florida Atlantic University  

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Venue：Ibaraki University  

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