The competition number of a graph having exactly one hole

AbstractLet D be an acyclic digraph. The competition graph of D has the same set of vertices as D and an edge between vertices u and v if and only if there is a vertex x in D such that (u,x) and (v,x) are arcs of D. The competition number of a graph G, denoted by k(G), is the smallest number k such that G together with k isolated vertices is the competition graph of an acyclic digraph. In this paper, we show that the competition number of a graph having exactly one chordless cycle of length at least 4 is at most two. We also give a large family of such graphs whose competition numbers are less than or equal to one

The competition number of a graph and the dimension of its hole space

The competition graph of a digraph D is a (simple undirected) graph which has the same vertex set as D and has an edge between x and y if and only if there exists a vertex v in D such that (x,v) and (y,v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of G is the smallest number of such isolated vertices. In general, it is hard to compute the competition number k(G) for a graph G and it has been one of important research problems in the study of competition graphs to characterize a graph by its competition number. Recently, the relationship between the competition number and the number of holes of a graph is being studied. A hole of a graph is a cycle of length at least 4 as an induced subgraph. In this paper, we conjecture that the dimension of the hole space of a graph is no smaller than the competition number of the graph. We verify this conjecture for various kinds of graphs and show that our conjectured inequality is indeed an equality for connected triangle-free graphs.Comment: 6 pages, 3 figure

Competitively tight graphs

The competition graph of a digraph $D$ is a (simple undirected) graph which has the same vertex set as $D$ and has an edge between two distinct vertices $x$ and $y$ if and only if there exists a vertex $v$ in $D$ such that $(x,v)$ and $(y,v)$ are arcs of $D$. For any graph $G$, $G$ together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number $k(G)$ of a graph $G$ is the smallest number of such isolated vertices. Computing the competition number of a graph is an NP-hard problem in general and has been one of the important research problems in the study of competition graphs. Opsut [1982] showed that the competition number of a graph $G$ is related to the edge clique cover number $\theta_E(G)$ of the graph $G$ via $\theta_E(G)-|V(G)|+2 \leq k(G) \leq \theta_E(G)$. We first show that for any positive integer $m$ satisfying $2 \leq m \leq |V(G)|$, there exists a graph $G$ with $k(G)=\theta_E(G)-|V(G)|+m$ and characterize a graph $G$ satisfying $k(G)=\theta_E(G)$. We then focus on what we call \emph{competitively tight graphs} $G$ which satisfy the lower bound, i.e., $k(G)=\theta_E(G)-|V(G)|+2$. We completely characterize the competitively tight graphs having at most two triangles. In addition, we provide a new upper bound for the competition number of a graph from which we derive a sufficient condition and a necessary condition for a graph to be competitively tight.Comment: 10 pages, 2 figure

The competition numbers of Hamming graphs with diameter at most three

The competition graph of a digraph D is a graph which has the same vertex set as D and has an edge between x and y if and only if there exists a vertex v in D such that (x,v) and (y,v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of a graph G is defined to be the smallest number of such isolated vertices. In general, it is hard to compute the competition number k(G) for a graph G and it has been one of important research problems in the study of competition graphs. In this paper, we compute the competition numbers of Hamming graphs with diameter at most three.Comment: 12 pages, 1 figur

Structure and Problem Hardness: Goal Asymmetry and DPLL Proofs in<br> SAT-Based Planning

In Verification and in (optimal) AI Planning, a successful method is to formulate the application as boolean satisfiability (SAT), and solve it with state-of-the-art DPLL-based procedures. There is a lack of understanding of why this works so well. Focussing on the Planning context, we identify a form of problem structure concerned with the symmetrical or asymmetrical nature of the cost of achieving the individual planning goals. We quantify this sort of structure with a simple numeric parameter called AsymRatio, ranging between 0 and 1. We run experiments in 10 benchmark domains from the International Planning Competitions since 2000; we show that AsymRatio is a good indicator of SAT solver performance in 8 of these domains. We then examine carefully crafted synthetic planning domains that allow control of the amount of structure, and that are clean enough for a rigorous analysis of the combinatorial search space. The domains are parameterized by size, and by the amount of structure. The CNFs we examine are unsatisfiable, encoding one planning step less than the length of the optimal plan. We prove upper and lower bounds on the size of the best possible DPLL refutations, under different settings of the amount of structure, as a function of size. We also identify the best possible sets of branching variables (backdoors). With minimum AsymRatio, we prove exponential lower bounds, and identify minimal backdoors of size linear in the number of variables. With maximum AsymRatio, we identify logarithmic DPLL refutations (and backdoors), showing a doubly exponential gap between the two structural extreme cases. The reasons for this behavior -- the proof arguments -- illuminate the prototypical patterns of structure causing the empirical behavior observed in the competition benchmarks