{
  "id": "1006.2631",
  "version": "v2",
  "published": "2010-06-14T08:25:46.000Z",
  "updated": "2010-09-29T17:07:02.000Z",
  "title": "The competition-common enemy graphs of digraphs satisfying Conditions $C(p)$ and $C'(p)$",
  "authors": [
    "Yoshio Sano"
  ],
  "comment": "8 pages, 2 figures",
  "journal": "Congressus Numerantium 202 (2010) 187-194",
  "categories": [
    "math.CO"
  ],
  "abstract": "S. -R. Kim and F. S. Roberts (2002) introduced the following conditions $C(p)$ and $C'(p)$ for digraphs as generalizations of the condition for digraphs to be semiorders. The condition $C(p)$ (resp. $C'(p)$) is: For any set $S$ of $p$ vertices in $D$, there exists $x \\in S$ such that $N^+_D(x) \\subseteq N^+_D(y)$ (resp. $N^-_D(x) \\subseteq N^-_D(y)$) for all $y \\in S$, where $N^+_D(x)$ (resp. $N^-_D(x)$) is the set of out-neighbors (resp. in-neighbors) of $x$ in $D$. The competition graph of a digraph $D$ is the (simple undirected) graph which has the same vertex set as $D$ and has an edge between two distinct vertices $x$ and $y$ if $N^+_D(x) \\cap N^+_D(y) \\neq \\emptyset$. Kim and Roberts characterized the competition graphs of digraphs which satisfy Condition $C(p)$. The competition-common enemy graph of a digraph $D$ is the graph which has the same vertex set as $D$ and has an edge between two distinct vertices $x$ and $y$ if it holds that both $N^+_D(x) \\cap N^+_D(y) \\neq \\emptyset$ and $N^-_D(x) \\cap N^-_D(y) \\neq \\emptyset$. In this note, we characterize the competition-common enemy graphs of digraphs satisfying Conditions $C(p)$ and $C'(p)$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-09-29T17:07:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C20",
      "06A06"
    ],
    "keywords": [
      "competition-common enemy graph",
      "digraphs satisfying conditions",
      "vertex set",
      "competition graph",
      "distinct vertices"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1006.2631S"
    }
  }
}