K_1,3-equipackable Graphs And K_1,3-equicoverable Graphs

Packing and covering problems are very important and fundamental in graph theory which have great significance in physics, computer network and combinatorics optimization research. Equipacking and equipcovering problems are dual problems.In this paper, we investigate K1,3-equipackable graphs and K1,3-equicoverable graphs. A graph G is called H-equipackable if its every maximal H-packing is also its maximum H-packing. A graph G is called H-equicoverable if its every minimal H-covering is also its minimum H-covering. In this paper, firstly, several special K1,3-equipackable caterpillar graphs are characterized, then by giving the characterization of K1,3-equipackable caterpillar graphs, K1,3-equipackable trees that don't have vertices of degree 3 are completely characterized, several special K1,3-equipackable graphs are characterized. Finally, several special K1,3- equicoverable trees are characterized, and the definition of cohering is given: G1 and G2 are the induced subgraphs of graphs G, satisfying G = G1 ? G2 and v = V(G1) ? V(G2), we call that G is got by cohering A with B at v. By cohering, K1,3-equicoverable trees are completely characterized.