The parameterized complexity of Strong Conflict-Free Vertex-Connection Colorability
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866911316085571584 |
|---|---|
| author | Feghali, Carl Le, Hoang-Oanh Le, Van Bang |
| author_facet | Feghali, Carl Le, Hoang-Oanh Le, Van Bang |
| contents | This paper continues the study of a new variant of graph coloring with a connectivity constraint recently introduced by Hsieh et al. [COCOON 2024]. A path in a vertex-colored graph is called conflict-free if there is a color that appears exactly once on its vertices. A connected graph is said to be strongly conflict-free vertex-connection $k$-colorable if it admits a (proper) vertex $k$-coloring such that any two distinct vertices are connected by a conflict-free shortest path. Among others, we show that deciding, for a given graph $G$ and an integer $k$, whether $G$ is strongly conflict-free $k$-colorable is fixed-parameter tractable when parameterized by the vertex cover number. But under the standard complexity-theoretic assumption NP $\not\subseteq$ coNP/poly, deciding, for a given graph $G$, whether $G$ is strongly conflict-free $3$-colorable does not admit a polynomial kernel, even for bipartite graphs. This kernel lower bound is in stark contrast to the ordinal $k$-Coloring problem which is known to admit a polynomial kernel when parameterized by the vertex cover number. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_11725 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The parameterized complexity of Strong Conflict-Free Vertex-Connection Colorability Feghali, Carl Le, Hoang-Oanh Le, Van Bang Data Structures and Algorithms Computational Complexity Discrete Mathematics Combinatorics This paper continues the study of a new variant of graph coloring with a connectivity constraint recently introduced by Hsieh et al. [COCOON 2024]. A path in a vertex-colored graph is called conflict-free if there is a color that appears exactly once on its vertices. A connected graph is said to be strongly conflict-free vertex-connection $k$-colorable if it admits a (proper) vertex $k$-coloring such that any two distinct vertices are connected by a conflict-free shortest path. Among others, we show that deciding, for a given graph $G$ and an integer $k$, whether $G$ is strongly conflict-free $k$-colorable is fixed-parameter tractable when parameterized by the vertex cover number. But under the standard complexity-theoretic assumption NP $\not\subseteq$ coNP/poly, deciding, for a given graph $G$, whether $G$ is strongly conflict-free $3$-colorable does not admit a polynomial kernel, even for bipartite graphs. This kernel lower bound is in stark contrast to the ordinal $k$-Coloring problem which is known to admit a polynomial kernel when parameterized by the vertex cover number. |
| title | The parameterized complexity of Strong Conflict-Free Vertex-Connection Colorability |
| topic | Data Structures and Algorithms Computational Complexity Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2512.11725 |