In a geometric drawing of Kn, trivially each 3-cycle bounds convex region: if two vertices are in that region, then so is the (geometric) edge between them. We define topological D Kn to be closed region R such any have (topological) them contained R.While drawings generalize drawings, they specialize ones. Therefore it might surprising all optimal (that is, crossing-minimal) were convex. Howev...

A set of vertices $S$ in a connected graph $G$ is a different-distance set if, for any vertex $w$ outside $S$, no two vertices in $S$ have the same distance to $w$.The lower and upper different-distance number of a graph are the order of a smallest, respectively largest, maximal different-distance set.We prove that a different-distance set induces either a special type of path or an independent...

A set $S subseteq V(G)$ is a semitotal dominating set of a graph $G$ if it is a dominating set of $G$ andevery vertex in $S$ is within distance 2 of another vertex of $S$. Thesemitotal domination number $gamma_{t2}(G)$ is the minimumcardinality of a semitotal dominating set of $G$.We show that the semitotal domination problem isAPX-complete for bounded-degree graphs, and the semitotal dominatio...

In this paper we initialize the study of independent domination in directed graphs. We show that an independent dominating set of an orientation of a graph is also an independent dominating set of the underlying graph, but that the converse is not true in general. We then prove existence and uniqueness theorems for several classes of digraphs including orientations of complete graphs, paths, tr...

Why are n-player games much more complex than two-player games? Is it much more difficult to cooperate or to compete? N -player Hackenbush is an n-player version of Blue-Red Hackenbush, a classic two-player combinatorial game played on graphs. Because of queer games, i.e., games where no player has a winning strategy, cooperation is a key-factor in n-player games and, as a consequence, nplayer ...

We investigate the combinatorial game Slime Trail.This game is played on a graph with a starting piece in a node. Each player's objective is to reach one of their own goal nodes. Every turn the current player moves the piece and deletes the node they came from. We show that the game is PSPACE-complete when played on a planar graph.

We show that the following (equivalent) problems are P-complete: 1. Does a given connuent NCE graph grammar only generate graphs of bounded tree-width? and 2. is the graph language generated by a given connuent NCE graph grammar an HR language? This settles the complexity of these important problems on graph grammars .

Let $G$ be a graph with vertex set $V(G)$. For any integer $kge 1$, a signed (total) $k$-dominating functionis a function $f: V(G) rightarrow { -1, 1}$ satisfying $sum_{xin N[v]}f(x)ge k$ ($sum_{xin N(v)}f(x)ge k$)for every $vin V(G)$, where $N(v)$ is the neighborhood of $v$ and $N[v]=N(v)cup{v}$. The minimum of the values$sum_{vin V(G)}f(v)$, taken over all signed (total) $k$-dominating functi...

In this paper we construct antimagic labelings of the regular complete multipartite graphs and we also extend the construction to some families of regular graphs.