Competition-common enemy graphs of degree-bounded digraphs
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| Format: | Preprint |
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2024
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| _version_ | 1866913359466594304 |
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| author | Choi, Myungho Chu, Hojin Kim, Suh-Ryung |
| author_facet | Choi, Myungho Chu, Hojin Kim, Suh-Ryung |
| contents | The competition-common enemy graph (CCE graph) of a digraph $D$ is the graph with the vertex set $V(D)$ and an edge $uv$ if and only if $u$ and $v$ have a common predator and a common prey in $D$. If each vertex of a digraph $D$ has indegree at most $i$ and outdegree at most $j$, then $D$ is called an $\langle i,j \rangle$ digraph. In this paper, we fully characterize the CCE graphs of $\langle 2,2\rangle$ digraphs. Then we investigate the CCE graphs of acyclic $\langle 2,2 \rangle$ digraphs, and prove that any CCE graph of an acyclic $\langle 2,2 \rangle$ digraph with at most seven components is interval, and the bound is sharp. While characterizing acyclic $\langle 2,2 \rangle$ digraphs that have interval graphs as their competition graphs, Hefner~{\it et al}. (1991) initiated the study of competition graphs of degree-bounded digraphs. Recently, Lee~{\em et al}. (2017) and Eoh and Kim (2021) studied phylogeny graphs of degree-bounded digraphs to extend their work. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_13363 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Competition-common enemy graphs of degree-bounded digraphs Choi, Myungho Chu, Hojin Kim, Suh-Ryung Combinatorics 05C20, 05C75 The competition-common enemy graph (CCE graph) of a digraph $D$ is the graph with the vertex set $V(D)$ and an edge $uv$ if and only if $u$ and $v$ have a common predator and a common prey in $D$. If each vertex of a digraph $D$ has indegree at most $i$ and outdegree at most $j$, then $D$ is called an $\langle i,j \rangle$ digraph. In this paper, we fully characterize the CCE graphs of $\langle 2,2\rangle$ digraphs. Then we investigate the CCE graphs of acyclic $\langle 2,2 \rangle$ digraphs, and prove that any CCE graph of an acyclic $\langle 2,2 \rangle$ digraph with at most seven components is interval, and the bound is sharp. While characterizing acyclic $\langle 2,2 \rangle$ digraphs that have interval graphs as their competition graphs, Hefner~{\it et al}. (1991) initiated the study of competition graphs of degree-bounded digraphs. Recently, Lee~{\em et al}. (2017) and Eoh and Kim (2021) studied phylogeny graphs of degree-bounded digraphs to extend their work. |
| title | Competition-common enemy graphs of degree-bounded digraphs |
| topic | Combinatorics 05C20, 05C75 |
| url | https://arxiv.org/abs/2405.13363 |