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WALCOM2026

Thu, 01 Jan 2026 00:00:00 +0000

(Induced) Subgraph Isomorphism and Maximum Common (Induced) Subgraph are fundamental problems in graph pattern matching and similarity computation. In graphs derived from time-series data or protein structures, a natural total ordering of vertices often arises from their underlying structure, such as temporal sequences or amino acid sequences. This motivates the study of problem variants that respect this inherent ordering. This paper addresses Ordered (Induced) Subgraph Isomorphism (O(I)SI) and its generalization, Maximum Common Ordered (Induced) Subgraph (MCO(I)S), which seek to find subgraph isomorphisms that preserve the vertex orderings of two given ordered graphs. Our main contributions are threefold: (1) We prove that these problems remain NP-complete even when restricted to small graph classes, such as trees of depth 2 and threshold graphs. (2) We establish a gap in computational complexity between OSI and OISI on certain graph classes. For instance, OSI is polynomial-time solvable for interval graphs with their interval orderings, whereas OISI remains NP-complete under the same setting. (3) We demonstrate that the tractability of these problems can depend on the vertex ordering. For example, while OISI is NP-complete on threshold graphs, its generalization, MCOIS, can be solved in polynomial time if the specific vertex orderings that characterize the threshold graphs are provided.

WALCOM2023-Kumagai

Wed, 01 Mar 2023 00:00:00 +0000

Position heaps are index structures of text strings used for the string matching problem. They are rooted trees whose edges and nodes are labeled and numbered, respectively. This paper is concerned with variants of the inverse problem of position heap construction and gives linear-time algorithms for those problems. The basic problem is to restore a text string from a rooted tree with labeled edges and numbered nodes. In the variant problems, the input trees may miss edge labels or node numbers which we must restore as well.

WALCOM2021

Mon, 01 Mar 2021 00:00:00 +0000

For a graph class $\mathcal{C}$, the $\mathcal{C}$-Edge-Deleion problem asks for a given graph $G$ to delete the minimum number of edges from $G$ in order to obtain a graph in $\mathcal{C}$. We study the $\mathcal{C}$-Edge-Deletion problem for $\mathcal{C}$ the class of interval graphs and other related graph classes. It follows from Courcelle’s Theorem that these problems are fixed parameter tractable when parameterized by treewidth. In this paper, we present concrete FPT algorithms for these problems. By giving explicit algorithms and analyzing these in detail, we obtain algorithms that are significantly faster than the algorithms obtained by using Courcelle’s theorem.

WALCOM2017

Wed, 29 Mar 2017 00:00:00 +0000

The token swapping problem (TSP) and its colored version are reconfiguration problems on graphs. This paper is concerned with the complexity of the TSP and two new variants; namely parallel TSP and parallel colored TSP. For a given graph where each vertex has a unique token on it, the TSP requires to find a shortest way to modify a token placement into another by swapping tokens on adjacent vertices. In the colored version, vertices and tokens are colored and the goal is to relocate tokens so that each vertex has a token of the same color. Their parallel versions allow simultaneous swaps on non-incident edges in one step. We investigate the time complexity of several restricted cases of those problems and show when those problems become tractable and remain intractable.