TY - GEN

T1 - Path-bicolorable graphs (Extended Abstract)

AU - Brandstädt, Andreas

AU - Golumbic, Martin C.

AU - Le, Van Bang

AU - Lipshteyn, Marina

PY - 2009

Y1 - 2009

N2 - In this paper, we introduce the notion of path-bicolorability that generalizes bipartite graphs in a natural way: For k ≥ 2, a graph G = (V,E) is P k -bicolorable if its vertex set V can be partitioned into two subsets (i.e., colors) V 1 and V 2 such that for every induced P k (i.e., path with exactly k - 1 edges and k vertices) in G, the two colors alternate along the P k , i.e., no two consecutive vertices of the P k belong to the same color V i , i = 1,2. Obviously, a graph is bipartite if and only if is P 2- bicolorable, every graph is P k -bicolorable for some k and if G is P k -bicolorable then it is P k + 1-bicolorable. The notion of P k -bicolorable graphs is motivated by a similar notion of cycle-bicolorable graphs introduced in connection with chordal probe graphs. Moreover, P 3- and P 4-bicolorable graphs are closely related to various other concepts such as 2-subcolorable graphs, P 4-bipartite graphs and alternately orientable graphs. We give a structural characterization of P 3-bicolorable graphs which also implies linear time recognition of these graphs. Moreover, we give a characterization of P 4-bicolorable graphs in terms of forbidden subgraphs.

AB - In this paper, we introduce the notion of path-bicolorability that generalizes bipartite graphs in a natural way: For k ≥ 2, a graph G = (V,E) is P k -bicolorable if its vertex set V can be partitioned into two subsets (i.e., colors) V 1 and V 2 such that for every induced P k (i.e., path with exactly k - 1 edges and k vertices) in G, the two colors alternate along the P k , i.e., no two consecutive vertices of the P k belong to the same color V i , i = 1,2. Obviously, a graph is bipartite if and only if is P 2- bicolorable, every graph is P k -bicolorable for some k and if G is P k -bicolorable then it is P k + 1-bicolorable. The notion of P k -bicolorable graphs is motivated by a similar notion of cycle-bicolorable graphs introduced in connection with chordal probe graphs. Moreover, P 3- and P 4-bicolorable graphs are closely related to various other concepts such as 2-subcolorable graphs, P 4-bipartite graphs and alternately orientable graphs. We give a structural characterization of P 3-bicolorable graphs which also implies linear time recognition of these graphs. Moreover, we give a characterization of P 4-bicolorable graphs in terms of forbidden subgraphs.

KW - Bipartite graphs

KW - Linear time recognition

KW - P -bicolorable graphs

KW - P -bicolorable graphs

UR - http://www.scopus.com/inward/record.url?scp=70249093280&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-02029-2_17

DO - 10.1007/978-3-642-02029-2_17

M3 - Conference contribution

AN - SCOPUS:70249093280

SN - 3642020283

SN - 9783642020285

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 172

EP - 182

BT - Graph Theory, Computational Intelligence and Thought - Essays Dedicated to Martin Charles Golumbic on the Occasion of His 60th Birthday

A2 - Lipshteyn, Marina

A2 - Levit, Vadim E.

A2 - McConnell, Ross M.

ER -