{
  "id": "1408.3204",
  "version": "v1",
  "published": "2014-08-14T07:12:54.000Z",
  "updated": "2014-08-14T07:12:54.000Z",
  "title": "Degree Monotone Paths and Graph Operations",
  "authors": [
    "Yair Caro",
    "Josef Lauri",
    "Christina Zarb"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A path $P$ in a graph $G$ is said to be a degree monotone path if the sequence of degrees of the vertices of $P$ in the order in which they appear on $P$ is monotonic. The length of the longest degree monotone path in $G$ is denoted by $mp(G)$. This parameter was first studied in an earlier paper by the authors where bounds in terms of other parameters of $G$ were obtained. In this paper we concentrate on the study of how $mp(G)$ changes under various operations on $G$. We first consider how $mp(G)$ changes when an edge is deleted, added, contracted or subdivided. We similarly consider the effects of adding or deleting a vertex. We sometimes restrict our attention to particular classes of graphs. Finally we study $mp(G \\times H)$ in terms of $mp(G)$ and $mp(H)$ where $\\times$ is either the Cartesian product or the join of two graphs. In all these cases we give bounds on the parameter $mp$ of the modified graph in terms of the original graph or graphs and we show that all the bounds are sharp.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-14T07:12:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C07",
      "05C76",
      "05C38"
    ],
    "keywords": [
      "graph operations",
      "longest degree monotone path",
      "original graph",
      "cartesian product",
      "earlier paper"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.3204C"
    }
  }
}