TY - JOUR

T1 - Finding a path with two labels forbidden in group-labeled graphs

AU - Kawase, Yasushi

AU - Kobayashi, Yusuke

AU - Yamaguchi, Yutaro

N1 - Funding Information:
We deeply grateful to anonymous reviewers for their careful reading and giving a number of insightful comments. This work was supported by JSPS KAKENHI Grant Numbers 24106002 , 24700004 , and 26887014 , by JSPS Grant-in-Aid for JSPS Research Fellow Grant Number 13J02522 , by JST ERATO Grant Number JPMJER1201 , and by JST CREST Grant Number JPMJCR14D2 .

PY - 2020/7

Y1 - 2020/7

N2 - The parity of the length of paths and cycles is a classical and well-studied topic in graph theory and theoretical computer science. The parity constraints can be extended to label constraints in a group-labeled graph, which is a directed graph with each arc labeled by an element of a group. Recently, paths and cycles in group-labeled graphs have been investigated, such as packing non-zero paths and cycles, where “non-zero” means that the identity element is a unique forbidden label. In this paper, we present a solution to finding an s–t path with two labels forbidden in a group-labeled graph. This also leads to an elementary solution to finding a zero s–t path in a Z3-labeled graph, which is the first nontrivial case of finding a zero path. This situation in fact generalizes the 2-disjoint paths problem in undirected graphs, which also motivates us to consider that setting. More precisely, we provide a polynomial-time algorithm for testing whether there are at most two possible labels of s–t paths in a group-labeled graph or not, and finding s–t paths attaining at least three distinct labels if exist. The algorithm is based on a necessary and sufficient condition for a group-labeled graph to have exactly two possible labels of s–t paths, which is the main technical contribution of this paper.

AB - The parity of the length of paths and cycles is a classical and well-studied topic in graph theory and theoretical computer science. The parity constraints can be extended to label constraints in a group-labeled graph, which is a directed graph with each arc labeled by an element of a group. Recently, paths and cycles in group-labeled graphs have been investigated, such as packing non-zero paths and cycles, where “non-zero” means that the identity element is a unique forbidden label. In this paper, we present a solution to finding an s–t path with two labels forbidden in a group-labeled graph. This also leads to an elementary solution to finding a zero s–t path in a Z3-labeled graph, which is the first nontrivial case of finding a zero path. This situation in fact generalizes the 2-disjoint paths problem in undirected graphs, which also motivates us to consider that setting. More precisely, we provide a polynomial-time algorithm for testing whether there are at most two possible labels of s–t paths in a group-labeled graph or not, and finding s–t paths attaining at least three distinct labels if exist. The algorithm is based on a necessary and sufficient condition for a group-labeled graph to have exactly two possible labels of s–t paths, which is the main technical contribution of this paper.

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U2 - 10.1016/j.jctb.2019.12.001

DO - 10.1016/j.jctb.2019.12.001

M3 - Article

AN - SCOPUS:85076580418

VL - 143

SP - 65

EP - 122

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

ER -